YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.