YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 0#(0(1(4(5(x1:S))))) -> 0#(3(5(x1:S)))
 0#(0(1(4(5(x1:S))))) -> 0#(4(1(0(3(5(x1:S))))))
 0#(0(1(4(5(x1:S))))) -> 3#(5(x1:S))
 0#(1(3(3(4(x1:S))))) -> 0#(0(3(1(3(4(x1:S))))))
 0#(1(3(3(4(x1:S))))) -> 0#(3(1(3(4(x1:S)))))
 0#(1(3(3(4(x1:S))))) -> 3#(1(3(4(x1:S))))
 0#(1(3(4(x1:S)))) -> 0#(3(x1:S))
 0#(1(3(4(x1:S)))) -> 0#(4(1(0(3(x1:S)))))
 0#(1(3(4(x1:S)))) -> 0#(4(1(1(3(x1:S)))))
 0#(1(3(4(x1:S)))) -> 3#(x1:S)
 0#(1(2(3(4(x1:S))))) -> 0#(3(x1:S))
 0#(1(2(3(4(x1:S))))) -> 0#(4(1(0(3(x1:S)))))
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(1(4(0(2(x1:S))))) -> 0#(4(1(5(0(2(x1:S))))))
 0#(2(1(4(4(x1:S))))) -> 0#(4(1(2(4(3(x1:S))))))
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 0#(4(1(3(2(x1:S)))))
 0#(2(1(4(x1:S)))) -> 0#(4(1(2(3(x1:S)))))
 0#(2(1(4(x1:S)))) -> 3#(2(x1:S))
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 0#(4(5(2(5(3(x1:S))))))
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(1(0(3(2(x1:S))))))
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(1(3(2(x1:S))))) -> 3#(1(0(3(2(x1:S)))))
 3#(0(2(1(4(x1:S))))) -> 0#(4(1(3(2(x1:S)))))
 3#(0(2(1(4(x1:S))))) -> 3#(2(x1:S))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 0#(4(1(2(0(3(x1:S))))))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 0#(4(1(5(3(4(x1:S))))))
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 0#(4(1(5(5(3(x1:S))))))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 0#(4(1(2(4(3(x1:S))))))
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))

Problem 1: 

SCC Processor:
-> Pairs:
 0#(0(1(4(5(x1:S))))) -> 0#(3(5(x1:S)))
 0#(0(1(4(5(x1:S))))) -> 0#(4(1(0(3(5(x1:S))))))
 0#(0(1(4(5(x1:S))))) -> 3#(5(x1:S))
 0#(1(3(3(4(x1:S))))) -> 0#(0(3(1(3(4(x1:S))))))
 0#(1(3(3(4(x1:S))))) -> 0#(3(1(3(4(x1:S)))))
 0#(1(3(3(4(x1:S))))) -> 3#(1(3(4(x1:S))))
 0#(1(3(4(x1:S)))) -> 0#(3(x1:S))
 0#(1(3(4(x1:S)))) -> 0#(4(1(0(3(x1:S)))))
 0#(1(3(4(x1:S)))) -> 0#(4(1(1(3(x1:S)))))
 0#(1(3(4(x1:S)))) -> 3#(x1:S)
 0#(1(2(3(4(x1:S))))) -> 0#(3(x1:S))
 0#(1(2(3(4(x1:S))))) -> 0#(4(1(0(3(x1:S)))))
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(1(4(0(2(x1:S))))) -> 0#(4(1(5(0(2(x1:S))))))
 0#(2(1(4(4(x1:S))))) -> 0#(4(1(2(4(3(x1:S))))))
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 0#(4(1(3(2(x1:S)))))
 0#(2(1(4(x1:S)))) -> 0#(4(1(2(3(x1:S)))))
 0#(2(1(4(x1:S)))) -> 3#(2(x1:S))
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 0#(4(5(2(5(3(x1:S))))))
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(1(0(3(2(x1:S))))))
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(1(3(2(x1:S))))) -> 3#(1(0(3(2(x1:S)))))
 3#(0(2(1(4(x1:S))))) -> 0#(4(1(3(2(x1:S)))))
 3#(0(2(1(4(x1:S))))) -> 3#(2(x1:S))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 0#(4(1(2(0(3(x1:S))))))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 0#(4(1(5(3(4(x1:S))))))
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 0#(4(1(5(5(3(x1:S))))))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 0#(4(1(2(4(3(x1:S))))))
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(0(1(4(5(x1:S))))) -> 0#(3(5(x1:S)))
 0#(1(3(4(x1:S)))) -> 0#(3(x1:S))
 0#(1(3(4(x1:S)))) -> 3#(x1:S)
 0#(1(2(3(4(x1:S))))) -> 0#(3(x1:S))
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
->->-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(0(1(4(5(x1:S))))) -> 0#(3(5(x1:S)))
 0#(1(3(4(x1:S)))) -> 0#(3(x1:S))
 0#(1(3(4(x1:S)))) -> 3#(x1:S)
 0#(1(2(3(4(x1:S))))) -> 0#(3(x1:S))
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
-> Usable rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X
[3](X) = 0
[1](X) = 2
[2](X) = 2.X
[4](X) = 0
[5](X) = 2.X
[0#](X) = X + 2
[3#](X) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 0#(1(3(4(x1:S)))) -> 0#(3(x1:S))
 0#(1(3(4(x1:S)))) -> 3#(x1:S)
 0#(1(2(3(4(x1:S))))) -> 0#(3(x1:S))
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(1(3(4(x1:S)))) -> 0#(3(x1:S))
 0#(1(3(4(x1:S)))) -> 3#(x1:S)
 0#(1(2(3(4(x1:S))))) -> 0#(3(x1:S))
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
->->-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(1(3(4(x1:S)))) -> 0#(3(x1:S))
 0#(1(3(4(x1:S)))) -> 3#(x1:S)
 0#(1(2(3(4(x1:S))))) -> 0#(3(x1:S))
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
-> Usable rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 1
[3](X) = 1
[1](X) = 2
[2](X) = 1
[4](X) = 0
[5](X) = 1
[0#](X) = 2.X
[3#](X) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 0#(1(3(4(x1:S)))) -> 3#(x1:S)
 0#(1(2(3(4(x1:S))))) -> 0#(3(x1:S))
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(1(3(4(x1:S)))) -> 3#(x1:S)
 0#(1(2(3(4(x1:S))))) -> 0#(3(x1:S))
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
->->-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(1(3(4(x1:S)))) -> 3#(x1:S)
 0#(1(2(3(4(x1:S))))) -> 0#(3(x1:S))
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
-> Usable rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 1
[3](X) = 1
[1](X) = 2
[2](X) = 1
[4](X) = 1
[5](X) = 1
[0#](X) = X + 1
[3#](X) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 0#(1(2(3(4(x1:S))))) -> 0#(3(x1:S))
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(1(2(3(4(x1:S))))) -> 0#(3(x1:S))
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
->->-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(1(2(3(4(x1:S))))) -> 0#(3(x1:S))
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
-> Usable rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X
[3](X) = 0
[1](X) = 2
[2](X) = 2
[4](X) = 0
[5](X) = 0
[0#](X) = 2.X + 2
[3#](X) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
->->-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(1(2(3(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
-> Usable rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X
[3](X) = 1
[1](X) = 2.X + 2
[2](X) = 2
[4](X) = 1
[5](X) = 0
[0#](X) = 2.X
[3#](X) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
->->-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(2(1(4(4(x1:S))))) -> 3#(x1:S)
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
-> Usable rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 0
[3](X) = 0
[1](X) = 1
[2](X) = X
[4](X) = 0
[5](X) = 2.X
[0#](X) = X + 1
[3#](X) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
->->-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(2(1(4(x1:S)))) -> 3#(x1:S)
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
-> Usable rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X
[3](X) = 0
[1](X) = 2.X + 2
[2](X) = 2.X + 2
[4](X) = 0
[5](X) = 2.X
[0#](X) = X + 2
[3#](X) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
->->-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(2(4(3(5(x1:S))))) -> 3#(x1:S)
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
-> Usable rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X
[3](X) = 0
[1](X) = 2
[2](X) = 2
[4](X) = 0
[5](X) = 0
[0#](X) = 2.X + 1
[3#](X) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 3#(0(1(3(2(x1:S))))) -> 0#(3(2(x1:S)))
 3#(0(4(0(2(x1:S))))) -> 0#(3(x1:S))
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
->->-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))

Problem 1: 

Subterm Processor:
-> Pairs:
 3#(0(4(0(2(x1:S))))) -> 3#(x1:S)
 3#(4(0(1(4(x1:S))))) -> 3#(4(x1:S))
 3#(4(0(1(5(x1:S))))) -> 3#(x1:S)
 3#(4(1(2(4(x1:S))))) -> 3#(x1:S)
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Projection:
 pi(3#) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(0(1(4(5(x1:S))))) -> 0(4(1(0(3(5(x1:S))))))
 0(1(0(2(4(x1:S))))) -> 2(0(0(4(1(1(x1:S))))))
 0(1(0(2(x1:S)))) -> 0(0(3(1(2(x1:S)))))
 0(1(3(3(4(x1:S))))) -> 0(0(3(1(3(4(x1:S))))))
 0(1(3(4(x1:S)))) -> 0(4(1(0(3(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(3(1(x1:S)))))
 0(1(3(4(x1:S)))) -> 0(4(1(1(3(x1:S)))))
 0(1(2(3(4(x1:S))))) -> 2(0(4(1(0(3(x1:S))))))
 0(1(4(0(2(x1:S))))) -> 0(4(1(5(0(2(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 0(4(0(3(1(4(x1:S))))))
 0(1(4(3(4(x1:S))))) -> 3(0(4(1(5(4(x1:S))))))
 0(1(4(3(5(x1:S))))) -> 5(4(5(0(3(1(x1:S))))))
 0(1(4(1(5(x1:S))))) -> 2(5(0(4(1(1(x1:S))))))
 0(1(5(0(2(x1:S))))) -> 0(0(4(1(2(5(x1:S))))))
 0(1(5(1(4(x1:S))))) -> 4(5(0(3(1(1(x1:S))))))
 0(2(1(4(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 0(2(1(4(5(x1:S))))) -> 0(4(1(2(5(2(x1:S))))))
 0(2(1(4(x1:S)))) -> 0(4(1(3(2(x1:S)))))
 0(2(1(4(x1:S)))) -> 0(4(1(2(3(x1:S)))))
 0(2(1(4(x1:S)))) -> 2(0(4(1(4(x1:S)))))
 0(2(1(4(x1:S)))) -> 5(5(0(4(1(2(x1:S))))))
 0(2(1(5(4(x1:S))))) -> 5(0(2(0(4(1(x1:S))))))
 0(2(1(5(x1:S)))) -> 5(0(4(1(2(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(2(5(x1:S)))))
 0(2(2(4(x1:S)))) -> 0(4(2(5(2(x1:S)))))
 0(2(4(3(5(x1:S))))) -> 0(4(5(2(5(3(x1:S))))))
 0(2(4(1(5(x1:S))))) -> 5(0(4(1(5(2(x1:S))))))
 0(2(5(1(4(x1:S))))) -> 0(0(5(4(1(2(x1:S))))))
 3(0(1(3(2(x1:S))))) -> 0(3(1(0(3(2(x1:S))))))
 3(0(2(1(4(x1:S))))) -> 4(0(4(1(3(2(x1:S))))))
 3(0(2(1(5(x1:S))))) -> 5(3(2(0(4(1(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(0(4(0(2(x1:S))))) -> 0(4(1(2(0(3(x1:S))))))
 3(0(5(1(4(x1:S))))) -> 3(0(4(1(1(5(x1:S))))))
 3(0(5(1(5(x1:S))))) -> 0(4(1(3(5(5(x1:S))))))
 3(2(4(1(2(x1:S))))) -> 3(1(2(2(5(4(x1:S))))))
 3(2(4(1(5(x1:S))))) -> 3(1(4(5(2(5(x1:S))))))
 3(4(0(1(2(x1:S))))) -> 0(4(2(0(3(1(x1:S))))))
 3(4(0(1(4(x1:S))))) -> 0(4(1(5(3(4(x1:S))))))
 3(4(0(1(5(x1:S))))) -> 0(4(1(5(5(3(x1:S))))))
 3(4(0(2(4(x1:S))))) -> 0(3(4(0(4(2(x1:S))))))
 3(4(0(2(x1:S)))) -> 3(0(4(5(2(x1:S)))))
 3(4(0(2(x1:S)))) -> 3(5(0(4(2(x1:S)))))
 3(4(3(0(2(x1:S))))) -> 3(3(0(4(1(2(x1:S))))))
 3(4(1(3(5(x1:S))))) -> 4(3(0(3(1(5(x1:S))))))
 3(4(1(2(4(x1:S))))) -> 0(4(1(2(4(3(x1:S))))))
 3(4(5(0(2(x1:S))))) -> 0(3(0(4(2(5(x1:S))))))
 3(5(0(2(2(x1:S))))) -> 0(3(2(5(2(5(x1:S))))))
 3(5(2(1(4(x1:S))))) -> 3(5(1(0(4(2(x1:S))))))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.