YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
0(3(x1:S)) -> 5(3(x1:S))
2(4(x1:S)) -> 0(7(x1:S))
2(7(x1:S)) -> 1(8(x1:S))
2(8(1(x1:S))) -> 8(x1:S)
2(8(x1:S)) -> 4(x1:S)
2(8(x1:S)) -> 7(x1:S)
4(7(x1:S)) -> 1(3(x1:S))
4(x1:S) -> 5(2(3(x1:S)))
4(x1:S) -> 9(6(6(x1:S)))
5(2(6(x1:S))) -> 6(2(4(x1:S)))
5(9(x1:S)) -> 0(x1:S)
5(3(x1:S)) -> 6(0(x1:S))
6(2(x1:S)) -> 7(7(x1:S))
6(6(x1:S)) -> 3(x1:S)
6(9(x1:S)) -> 9(x1:S)
7(0(x1:S)) -> 9(3(x1:S))
7(2(x1:S)) -> 4(x1:S)
9(5(9(x1:S))) -> 5(7(x1:S))
9(7(x1:S)) -> 7(5(x1:S))
9(x1:S) -> 6(7(x1:S))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 2#(4(x1:S)) -> 0#(7(x1:S))
 2#(4(x1:S)) -> 7#(x1:S)
 2#(8(x1:S)) -> 4#(x1:S)
 2#(8(x1:S)) -> 7#(x1:S)
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))

Problem 1: 

SCC Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 2#(4(x1:S)) -> 0#(7(x1:S))
 2#(4(x1:S)) -> 7#(x1:S)
 2#(8(x1:S)) -> 4#(x1:S)
 2#(8(x1:S)) -> 7#(x1:S)
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 2#(4(x1:S)) -> 0#(7(x1:S))
 2#(4(x1:S)) -> 7#(x1:S)
 2#(8(x1:S)) -> 4#(x1:S)
 2#(8(x1:S)) -> 7#(x1:S)
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
->->-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 2#(4(x1:S)) -> 0#(7(x1:S))
 2#(4(x1:S)) -> 7#(x1:S)
 2#(8(x1:S)) -> 4#(x1:S)
 2#(8(x1:S)) -> 7#(x1:S)
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
-> Usable rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 0
[2](X) = 2.X
[4](X) = 0
[5](X) = 0
[6](X) = 0
[7](X) = 0
[9](X) = 0
[1](X) = 0
[3](X) = 0
[8](X) = 2
[0#](X) = 2
[2#](X) = 2.X + 2
[4#](X) = 2
[5#](X) = 2
[6#](X) = 2
[7#](X) = 2
[9#](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 2#(4(x1:S)) -> 0#(7(x1:S))
 2#(4(x1:S)) -> 7#(x1:S)
 2#(8(x1:S)) -> 7#(x1:S)
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 2#(4(x1:S)) -> 0#(7(x1:S))
 2#(4(x1:S)) -> 7#(x1:S)
 2#(8(x1:S)) -> 7#(x1:S)
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
->->-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 2#(4(x1:S)) -> 0#(7(x1:S))
 2#(4(x1:S)) -> 7#(x1:S)
 2#(8(x1:S)) -> 7#(x1:S)
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
-> Usable rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 1
[2](X) = 2
[4](X) = 1
[5](X) = 1
[6](X) = 1
[7](X) = 1
[9](X) = 1
[1](X) = 1
[3](X) = 1
[8](X) = 2
[0#](X) = 2
[2#](X) = X + 1
[4#](X) = 2
[5#](X) = 2
[6#](X) = 2
[7#](X) = 2
[9#](X) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 2#(4(x1:S)) -> 0#(7(x1:S))
 2#(4(x1:S)) -> 7#(x1:S)
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 2#(4(x1:S)) -> 0#(7(x1:S))
 2#(4(x1:S)) -> 7#(x1:S)
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
->->-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 2#(4(x1:S)) -> 0#(7(x1:S))
 2#(4(x1:S)) -> 7#(x1:S)
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
-> Usable rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 2
->Bound:
 1
->Interpretation:
 
[0](X) = 0
[2](X) = [1 0;0 1].X + [0;1]
[4](X) = [1;0]
[5](X) = [0 1;0 0].X
[6](X) = 0
[7](X) = [0 1;0 0].X
[9](X) = 0
[1](X) = [0 0;0 1].X
[3](X) = 0
[8](X) = [0 1;0 0].X + [1;1]
[0#](X) = [1;1]
[2#](X) = [1 1;0 1].X + [1;1]
[4#](X) = [1;1]
[5#](X) = [0 1;0 0].X + [1;1]
[6#](X) = [1;1]
[7#](X) = [1;1]
[9#](X) = [1 0;0 0].X + [1;1]

Problem 1: 

SCC Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 2#(4(x1:S)) -> 7#(x1:S)
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 2#(4(x1:S)) -> 7#(x1:S)
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
->->-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 2#(4(x1:S)) -> 7#(x1:S)
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
-> Usable rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 2
->Bound:
 1
->Interpretation:
 
[0](X) = 0
[2](X) = [0 0;0 1].X + [1;0]
[4](X) = [0;1]
[5](X) = 0
[6](X) = 0
[7](X) = [0 0;1 0].X
[9](X) = 0
[1](X) = [1 0;0 0].X
[3](X) = 0
[8](X) = [0 0;1 0].X + [1;1]
[0#](X) = [1;1]
[2#](X) = [1 1;0 1].X + [1;1]
[4#](X) = [1;1]
[5#](X) = [1 0;1 0].X + [1;1]
[6#](X) = [1;1]
[7#](X) = [1;1]
[9#](X) = [0 1;1 1].X + [1;1]

Problem 1: 

SCC Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 2#(4(x1:S))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
->->-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 4#(x1:S) -> 6#(6(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
-> Usable rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 2
->Bound:
 1
->Interpretation:
 
[0](X) = [1;0]
[2](X) = [1 1;1 0].X + [0;1]
[4](X) = [1 0;0 0].X + [1;0]
[5](X) = [1 0;0 0].X + [1;0]
[6](X) = [1 0;0 0].X
[7](X) = [1 1;0 0].X
[9](X) = [1 1;0 0].X + [1;0]
[1](X) = [0 1;1 1].X
[3](X) = 0
[8](X) = [1 1;0 0].X + [1;0]
[0#](X) = [1;1]
[4#](X) = [1 0;0 0].X + [1;1]
[5#](X) = [1 1;0 0].X + [1;1]
[6#](X) = [1 0;0 0].X + [0;1]
[7#](X) = [1 1;0 0].X + [0;1]
[9#](X) = [1 1;0 0].X + [1;1]

Problem 1: 

SCC Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
->->-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 4#(x1:S) -> 6#(x1:S)
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
-> Usable rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 2
->Bound:
 1
->Interpretation:
 
[0](X) = [1;0]
[2](X) = [1 1;1 1].X + [0;1]
[4](X) = [0 0;0 1].X + [1;1]
[5](X) = [0 0;1 1].X + [1;0]
[6](X) = [0 0;0 1].X + [1;0]
[7](X) = [0 0;1 1].X + [1;0]
[9](X) = [0 0;1 1].X + [1;0]
[1](X) = [1 1;1 0].X + [0;1]
[3](X) = 0
[8](X) = [1 1;0 0].X + [0;1]
[0#](X) = [1;1]
[4#](X) = [0 1;0 1].X + [1;1]
[5#](X) = [1 1;0 1].X + [1;1]
[6#](X) = [0 1;0 1].X
[7#](X) = [0 1;1 1].X
[9#](X) = [1 1;1 1].X

Problem 1: 

SCC Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
->->-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 6#(0(x1:S))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
-> Usable rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 0
[2](X) = 2.X + 1
[4](X) = 0
[5](X) = 0
[6](X) = 0
[7](X) = 0
[9](X) = 0
[1](X) = 0
[3](X) = 0
[8](X) = 1
[0#](X) = 2
[4#](X) = 2
[5#](X) = 2
[6#](X) = 2.X
[7#](X) = 2
[9#](X) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 5#(9(x1:S)) -> 0#(x1:S)
 5#(3(x1:S)) -> 0#(x1:S)
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 5#(3(x1:S)) -> 0#(x1:S)
->->-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->->Cycle:
->->-> Pairs:
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
->->-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
 5#(3(x1:S)) -> 0#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Projection:
 pi(0#) = 1
 pi(5#) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 0#(3(x1:S)) -> 5#(3(x1:S))
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 6#(7(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
-> Usable rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 1
[2](X) = 2
[4](X) = 1
[5](X) = 1
[6](X) = 1
[7](X) = 1
[9](X) = 1
[1](X) = 1
[3](X) = 1
[8](X) = 0
[4#](X) = 2
[5#](X) = 2
[6#](X) = X
[7#](X) = 2
[9#](X) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 7#(x1:S)
->->-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 4#(x1:S) -> 9#(6(6(x1:S)))
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
-> Usable rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 2
->Bound:
 1
->Interpretation:
 
[0](X) = [0;1]
[2](X) = [1 1;0 1].X + [1;0]
[4](X) = [1 0;0 0].X + [1;1]
[5](X) = [1 1;0 0].X + [0;1]
[6](X) = [1 0;0 0].X + [0;1]
[7](X) = [1 1;0 0].X + [0;1]
[9](X) = [1 1;0 0].X + [0;1]
[1](X) = [1 1;0 0].X + [1;1]
[3](X) = 0
[8](X) = [0 1;1 0].X + [0;1]
[4#](X) = [1 0;1 0].X + [1;1]
[5#](X) = [1 1;1 1].X + [0;1]
[6#](X) = [1 0;1 1].X
[7#](X) = [1 0;1 0].X + [0;1]
[9#](X) = [1 0;1 0].X + [0;1]

Problem 1.2: 

SCC Processor:
-> Pairs:
 5#(2(6(x1:S))) -> 4#(x1:S)
 5#(2(6(x1:S))) -> 6#(2(4(x1:S)))
 6#(2(x1:S)) -> 7#(7(x1:S))
 6#(2(x1:S)) -> 7#(x1:S)
 7#(0(x1:S)) -> 9#(3(x1:S))
 7#(2(x1:S)) -> 4#(x1:S)
 9#(5(9(x1:S))) -> 5#(7(x1:S))
 9#(5(9(x1:S))) -> 7#(x1:S)
 9#(7(x1:S)) -> 5#(x1:S)
 9#(7(x1:S)) -> 7#(5(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 7#(0(x1:S)) -> 9#(3(x1:S))
 9#(x1:S) -> 7#(x1:S)
->->-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 7#(0(x1:S)) -> 9#(3(x1:S))
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[3](X) = 2.X + 1
[7#](X) = 2.X + 2
[9#](X) = 2.X + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 9#(x1:S) -> 7#(x1:S)
-> Rules:
 0(3(x1:S)) -> 5(3(x1:S))
 2(4(x1:S)) -> 0(7(x1:S))
 2(7(x1:S)) -> 1(8(x1:S))
 2(8(1(x1:S))) -> 8(x1:S)
 2(8(x1:S)) -> 4(x1:S)
 2(8(x1:S)) -> 7(x1:S)
 4(7(x1:S)) -> 1(3(x1:S))
 4(x1:S) -> 5(2(3(x1:S)))
 4(x1:S) -> 9(6(6(x1:S)))
 5(2(6(x1:S))) -> 6(2(4(x1:S)))
 5(9(x1:S)) -> 0(x1:S)
 5(3(x1:S)) -> 6(0(x1:S))
 6(2(x1:S)) -> 7(7(x1:S))
 6(6(x1:S)) -> 3(x1:S)
 6(9(x1:S)) -> 9(x1:S)
 7(0(x1:S)) -> 9(3(x1:S))
 7(2(x1:S)) -> 4(x1:S)
 9(5(9(x1:S))) -> 5(7(x1:S))
 9(7(x1:S)) -> 7(5(x1:S))
 9(x1:S) -> 6(7(x1:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.