YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.