YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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


The problem is decomposed in 2 subproblems.

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

The problem is finite.

Problem 1.2.2: 

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

Problem 1.2.2: 

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

Problem 1.2.2: 

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

Problem 1.2.2: 

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

The problem is finite.