YES Problem 1: (VAR v_NonEmpty:S x1:S) (RULES 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ) Problem 1: Dependency Pairs Processor: -> Pairs: 1#(2(5(x1:S))) -> 1#(2(2(1(0(1(x1:S)))))) 1#(2(5(x1:S))) -> 1#(0(1(x1:S))) 1#(2(5(x1:S))) -> 1#(0(5(0(5(4(x1:S)))))) 1#(2(5(x1:S))) -> 1#(x1:S) 1#(2(5(x1:S))) -> 2#(1(0(1(x1:S)))) 1#(2(5(x1:S))) -> 2#(2(1(0(1(x1:S))))) 1#(2(5(x1:S))) -> 4#(x1:S) 1#(2(5(x1:S))) -> 5#(4(x1:S)) 1#(2(5(x1:S))) -> 5#(0(5(4(x1:S)))) 1#(4(5(x1:S))) -> 1#(2(4(0(2(1(x1:S)))))) 1#(4(5(x1:S))) -> 1#(x1:S) 1#(4(5(x1:S))) -> 2#(1(x1:S)) 1#(4(5(x1:S))) -> 2#(4(0(2(1(x1:S))))) 1#(4(5(x1:S))) -> 4#(0(2(1(x1:S)))) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 4#(0(2(2(3(3(x1:S)))))) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 2#(0(4(1(3(3(x1:S)))))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(0(1(x1:S))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(4(x1:S))) -> 2#(0(5(1(0(1(x1:S)))))) 2#(5(4(x1:S))) -> 5#(1(0(1(x1:S)))) 2#(5(x1:S)) -> 1#(x1:S) 2#(5(x1:S)) -> 2#(2(0(5(0(1(x1:S)))))) 2#(5(x1:S)) -> 2#(0(5(0(1(x1:S))))) 2#(5(x1:S)) -> 5#(0(1(x1:S))) 3#(2(5(x1:S))) -> 1#(0(5(x1:S))) 3#(2(5(x1:S))) -> 2#(0(1(0(5(x1:S))))) 3#(2(5(x1:S))) -> 3#(2(0(1(0(5(x1:S)))))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(4(2(x1:S))) -> 3#(4(0(2(2(2(x1:S)))))) 3#(4(2(x1:S))) -> 4#(0(2(2(2(x1:S))))) 3#(5(1(x1:S))) -> 1#(4(1(0(1(x1:S))))) 3#(5(1(x1:S))) -> 1#(0(1(x1:S))) 3#(5(1(x1:S))) -> 2#(1(4(1(0(1(x1:S)))))) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 2#(0(0(5(x1:S)))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(1(0(1(x1:S)))) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(2(0(0(5(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(3(4(0(4(2(x1:S)))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 3#(4(0(4(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(3(x1:S))) -> 4#(0(4(2(x1:S)))) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(4(x1:S))) -> 5#(0(0(1(2(x1:S))))) 3#(5(5(x1:S))) -> 1#(0(5(x1:S))) 3#(5(5(x1:S))) -> 4#(1(0(5(x1:S)))) 3#(5(5(x1:S))) -> 5#(4(1(0(5(x1:S))))) 3#(5(x1:S)) -> 1#(3(2(0(0(1(x1:S)))))) 3#(5(x1:S)) -> 1#(x1:S) 3#(5(x1:S)) -> 2#(0(0(1(x1:S)))) 3#(5(x1:S)) -> 3#(2(0(0(1(x1:S))))) 4#(5(1(x1:S))) -> 1#(0(5(3(3(x1:S))))) 4#(5(1(x1:S))) -> 2#(1(0(5(3(3(x1:S)))))) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 1#(0(0(4(x1:S)))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(2(x1:S))) -> 5#(1(0(0(4(x1:S))))) 4#(5(4(x1:S))) -> 1#(0(4(2(x1:S)))) 4#(5(4(x1:S))) -> 2#(1(0(4(2(x1:S))))) 4#(5(4(x1:S))) -> 2#(2(1(0(4(2(x1:S)))))) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 1#(0(1(2(2(x1:S))))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(3(x1:S))) -> 5#(1(0(1(2(2(x1:S)))))) 5#(5(4(x1:S))) -> 1#(0(4(2(2(x1:S))))) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) 5#(5(4(x1:S))) -> 5#(1(0(4(2(2(x1:S)))))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: SCC Processor: -> Pairs: 1#(2(5(x1:S))) -> 1#(2(2(1(0(1(x1:S)))))) 1#(2(5(x1:S))) -> 1#(0(1(x1:S))) 1#(2(5(x1:S))) -> 1#(0(5(0(5(4(x1:S)))))) 1#(2(5(x1:S))) -> 1#(x1:S) 1#(2(5(x1:S))) -> 2#(1(0(1(x1:S)))) 1#(2(5(x1:S))) -> 2#(2(1(0(1(x1:S))))) 1#(2(5(x1:S))) -> 4#(x1:S) 1#(2(5(x1:S))) -> 5#(4(x1:S)) 1#(2(5(x1:S))) -> 5#(0(5(4(x1:S)))) 1#(4(5(x1:S))) -> 1#(2(4(0(2(1(x1:S)))))) 1#(4(5(x1:S))) -> 1#(x1:S) 1#(4(5(x1:S))) -> 2#(1(x1:S)) 1#(4(5(x1:S))) -> 2#(4(0(2(1(x1:S))))) 1#(4(5(x1:S))) -> 4#(0(2(1(x1:S)))) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 4#(0(2(2(3(3(x1:S)))))) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 2#(0(4(1(3(3(x1:S)))))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(0(1(x1:S))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(4(x1:S))) -> 2#(0(5(1(0(1(x1:S)))))) 2#(5(4(x1:S))) -> 5#(1(0(1(x1:S)))) 2#(5(x1:S)) -> 1#(x1:S) 2#(5(x1:S)) -> 2#(2(0(5(0(1(x1:S)))))) 2#(5(x1:S)) -> 2#(0(5(0(1(x1:S))))) 2#(5(x1:S)) -> 5#(0(1(x1:S))) 3#(2(5(x1:S))) -> 1#(0(5(x1:S))) 3#(2(5(x1:S))) -> 2#(0(1(0(5(x1:S))))) 3#(2(5(x1:S))) -> 3#(2(0(1(0(5(x1:S)))))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(4(2(x1:S))) -> 3#(4(0(2(2(2(x1:S)))))) 3#(4(2(x1:S))) -> 4#(0(2(2(2(x1:S))))) 3#(5(1(x1:S))) -> 1#(4(1(0(1(x1:S))))) 3#(5(1(x1:S))) -> 1#(0(1(x1:S))) 3#(5(1(x1:S))) -> 2#(1(4(1(0(1(x1:S)))))) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 2#(0(0(5(x1:S)))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(1(0(1(x1:S)))) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(2(0(0(5(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(3(4(0(4(2(x1:S)))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 3#(4(0(4(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(3(x1:S))) -> 4#(0(4(2(x1:S)))) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(4(x1:S))) -> 5#(0(0(1(2(x1:S))))) 3#(5(5(x1:S))) -> 1#(0(5(x1:S))) 3#(5(5(x1:S))) -> 4#(1(0(5(x1:S)))) 3#(5(5(x1:S))) -> 5#(4(1(0(5(x1:S))))) 3#(5(x1:S)) -> 1#(3(2(0(0(1(x1:S)))))) 3#(5(x1:S)) -> 1#(x1:S) 3#(5(x1:S)) -> 2#(0(0(1(x1:S)))) 3#(5(x1:S)) -> 3#(2(0(0(1(x1:S))))) 4#(5(1(x1:S))) -> 1#(0(5(3(3(x1:S))))) 4#(5(1(x1:S))) -> 2#(1(0(5(3(3(x1:S)))))) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 1#(0(0(4(x1:S)))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(2(x1:S))) -> 5#(1(0(0(4(x1:S))))) 4#(5(4(x1:S))) -> 1#(0(4(2(x1:S)))) 4#(5(4(x1:S))) -> 2#(1(0(4(2(x1:S))))) 4#(5(4(x1:S))) -> 2#(2(1(0(4(2(x1:S)))))) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 1#(0(1(2(2(x1:S))))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(3(x1:S))) -> 5#(1(0(1(2(2(x1:S)))))) 5#(5(4(x1:S))) -> 1#(0(4(2(2(x1:S))))) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) 5#(5(4(x1:S))) -> 5#(1(0(4(2(2(x1:S)))))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(2(5(x1:S))) -> 1#(x1:S) 1#(2(5(x1:S))) -> 4#(x1:S) 1#(2(5(x1:S))) -> 5#(4(x1:S)) 1#(4(5(x1:S))) -> 1#(x1:S) 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(2(5(x1:S))) -> 1#(x1:S) 1#(2(5(x1:S))) -> 4#(x1:S) 1#(2(5(x1:S))) -> 5#(4(x1:S)) 1#(4(5(x1:S))) -> 1#(x1:S) 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = X [4](X) = 2.X + 1 [5](X) = 2.X + 2 [0](X) = 0 [1#](X) = X [2#](X) = 2.X [3#](X) = 2.X [4#](X) = 2.X + 1 [5#](X) = X + 1 Problem 1: SCC Processor: -> Pairs: 1#(2(5(x1:S))) -> 4#(x1:S) 1#(2(5(x1:S))) -> 5#(4(x1:S)) 1#(4(5(x1:S))) -> 1#(x1:S) 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(2(5(x1:S))) -> 4#(x1:S) 1#(2(5(x1:S))) -> 5#(4(x1:S)) 1#(4(5(x1:S))) -> 1#(x1:S) 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(2(5(x1:S))) -> 4#(x1:S) 1#(2(5(x1:S))) -> 5#(4(x1:S)) 1#(4(5(x1:S))) -> 1#(x1:S) 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = X [4](X) = X + 2 [5](X) = 2.X + 2 [0](X) = 0 [1#](X) = 2.X + 2 [2#](X) = X + 2 [3#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: 1#(2(5(x1:S))) -> 5#(4(x1:S)) 1#(4(5(x1:S))) -> 1#(x1:S) 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(2(5(x1:S))) -> 5#(4(x1:S)) 1#(4(5(x1:S))) -> 1#(x1:S) 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(2(5(x1:S))) -> 5#(4(x1:S)) 1#(4(5(x1:S))) -> 1#(x1:S) 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = X [4](X) = 2.X [5](X) = 2.X + 2 [0](X) = 0 [1#](X) = X + 1 [2#](X) = 2.X + 2 [3#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: 1#(4(5(x1:S))) -> 1#(x1:S) 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(4(5(x1:S))) -> 1#(x1:S) 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(4(5(x1:S))) -> 1#(x1:S) 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = X [4](X) = 2.X + 1 [5](X) = 2.X + 2 [0](X) = 0 [1#](X) = 2.X + 1 [2#](X) = 2.X + 2 [3#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(4(5(x1:S))) -> 2#(1(x1:S)) 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X [2](X) = X [3](X) = X + 1 [4](X) = X [5](X) = 2.X + 2 [0](X) = 0 [1#](X) = 2.X + 2 [2#](X) = 2.X [3#](X) = 2.X [4#](X) = 2.X [5#](X) = 2.X Problem 1: SCC Processor: -> Pairs: 2#(5(1(x1:S))) -> 1#(2(3(x1:S))) 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 1#(3(3(x1:S))) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 2#(5(4(x1:S))) -> 1#(x1:S) 2#(5(x1:S)) -> 1#(x1:S) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 1#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 1#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 3#(5(x1:S)) -> 1#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 1#(3(2(1(x1:S)))) 4#(5(x1:S)) -> 1#(x1:S) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 1#(2(2(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 2#(5(1(x1:S))) -> 2#(1(2(3(x1:S)))) 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X [2](X) = X [3](X) = X [4](X) = 2.X + 2 [5](X) = 2.X + 2 [0](X) = 0 [2#](X) = 2.X + 2 [3#](X) = 2.X + 2 [4#](X) = 2.X [5#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 2#(5(1(x1:S))) -> 2#(2(1(2(3(x1:S))))) 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = X [4](X) = X + 2 [5](X) = 2.X + 2 [0](X) = 0 [2#](X) = 2.X + 2 [3#](X) = 2.X [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 2#(5(1(x1:S))) -> 2#(2(2(1(2(3(x1:S)))))) 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = X [4](X) = X + 2 [5](X) = 2.X + 2 [0](X) = 0 [2#](X) = X + 2 [3#](X) = 2.X [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 2#(5(1(x1:S))) -> 2#(3(x1:S)) 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = X [4](X) = X + 2 [5](X) = 2.X + 2 [0](X) = 0 [2#](X) = 2.X + 1 [3#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = X Problem 1: SCC Processor: -> Pairs: 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 2#(5(1(x1:S))) -> 3#(x1:S) 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = X [4](X) = 2.X + 1 [5](X) = 2.X + 2 [0](X) = 0 [2#](X) = 2.X [3#](X) = 2.X + 1 [4#](X) = 2.X [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 2#(5(2(x1:S))) -> 2#(2(3(3(x1:S)))) 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X [2](X) = X [3](X) = X [4](X) = 2.X + 2 [5](X) = 2.X + 2 [0](X) = 0 [2#](X) = 2.X [3#](X) = 2.X [4#](X) = 2.X [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 2#(5(2(x1:S))) -> 2#(3(3(x1:S))) 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = X [4](X) = 2.X + 1 [5](X) = 2.X + 2 [0](X) = 0 [2#](X) = X [3#](X) = 2.X + 2 [4#](X) = 2.X [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 2#(5(2(x1:S))) -> 3#(3(x1:S)) 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = X [4](X) = 2.X + 1 [5](X) = 2.X + 2 [0](X) = 0 [2#](X) = 2.X + 2 [3#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 2#(5(2(x1:S))) -> 3#(x1:S) 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = X [4](X) = X + 2 [5](X) = 2.X + 2 [0](X) = 0 [2#](X) = 2.X + 2 [3#](X) = 2.X [4#](X) = 2.X + 2 [5#](X) = 2.X Problem 1: SCC Processor: -> Pairs: 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 2#(5(3(x1:S))) -> 3#(3(x1:S)) 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = X [4](X) = 2.X + 1 [5](X) = 2.X + 2 [0](X) = 0 [2#](X) = 2.X [3#](X) = 2.X [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 2#(5(3(x1:S))) -> 4#(1(3(3(x1:S)))) 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = X [4](X) = X + 2 [5](X) = 2.X + 2 [0](X) = 0 [2#](X) = 2.X + 2 [3#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X Problem 1: SCC Processor: -> Pairs: 3#(4(2(x1:S))) -> 2#(2(2(x1:S))) 3#(4(2(x1:S))) -> 2#(2(x1:S)) 3#(5(1(x1:S))) -> 2#(2(3(4(x1:S)))) 3#(5(1(x1:S))) -> 2#(3(4(x1:S))) 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 2#(1(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(2(x1:S))) 3#(5(2(x1:S))) -> 2#(2(x1:S)) 3#(5(2(x1:S))) -> 2#(3(3(2(1(2(x1:S)))))) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 2#(x1:S) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 3#(5(4(x1:S))) -> 2#(x1:S) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 2#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 2#(1(3(2(1(x1:S))))) 4#(5(x1:S)) -> 2#(1(x1:S)) 4#(5(x1:S)) -> 2#(2(1(3(2(1(x1:S)))))) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(3(x1:S))) -> 2#(2(x1:S)) 5#(5(3(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 2#(2(x1:S)) 5#(5(4(x1:S))) -> 2#(x1:S) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 3#(5(1(x1:S))) -> 3#(4(x1:S)) 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = 2.X [4](X) = X + 2 [5](X) = 2.X + 2 [0](X) = 0 [3#](X) = 2.X + 2 [4#](X) = 2.X + 1 [5#](X) = X + 1 Problem 1: SCC Processor: -> Pairs: 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 3#(5(1(x1:S))) -> 4#(2(2(3(4(x1:S))))) 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = 2.X [4](X) = X + 2 [5](X) = 2.X + 2 [0](X) = 0 [3#](X) = 2.X + 2 [4#](X) = 2.X [5#](X) = X Problem 1: SCC Processor: -> Pairs: 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 3#(5(1(x1:S))) -> 4#(x1:S) 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = 2.X [2](X) = X [3](X) = X [4](X) = 2.X [5](X) = 2.X + 1 [0](X) = 0 [3#](X) = X + 2 [4#](X) = X + 2 [5#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 3#(5(1(x1:S))) -> 5#(x1:S) 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X [2](X) = X [3](X) = X + 1 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [0](X) = 0 [3#](X) = 2.X + 1 [4#](X) = X + 2 [5#](X) = 2.X Problem 1: SCC Processor: -> Pairs: 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 3#(5(2(x1:S))) -> 3#(2(1(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = 2.X [4](X) = X + 2 [5](X) = 2.X + 2 [0](X) = 0 [3#](X) = 2.X + 1 [4#](X) = 2.X + 2 [5#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 3#(5(2(x1:S))) -> 3#(2(2(2(x1:S)))) 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = 2.X [4](X) = 2.X + 2 [5](X) = 2.X + 2 [0](X) = 0 [3#](X) = 2.X + 2 [4#](X) = 2.X + 1 [5#](X) = X Problem 1: SCC Processor: -> Pairs: 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 3#(5(2(x1:S))) -> 3#(3(2(1(2(x1:S))))) 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X [2](X) = X [3](X) = 2.X [4](X) = 2.X + 2 [5](X) = 2.X + 2 [0](X) = 0 [3#](X) = 2.X + 1 [4#](X) = 2.X + 1 [5#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 3#(5(2(x1:S))) -> 4#(3(2(2(2(x1:S))))) 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = 2.X [2](X) = X [3](X) = X [4](X) = 2.X + 1 [5](X) = 2.X + 2 [0](X) = 0 [3#](X) = 2.X + 2 [4#](X) = 2.X + 1 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 3#(5(3(x1:S))) -> 4#(2(x1:S)) 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X + 1 [2](X) = X [3](X) = 2.X [4](X) = 2.X + 2 [5](X) = 2.X + 2 [0](X) = 0 [3#](X) = X + 2 [4#](X) = 2.X [5#](X) = X Problem 1: SCC Processor: -> Pairs: 4#(5(1(x1:S))) -> 3#(3(x1:S)) 4#(5(1(x1:S))) -> 3#(x1:S) 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 4#(5(x1:S)) -> 3#(2(1(x1:S))) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 4#(5(1(x1:S))) -> 5#(3(3(x1:S))) 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = X [2](X) = X [3](X) = 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [0](X) = 0 [4#](X) = 2.X + 1 [5#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) 5#(5(4(x1:S))) -> 4#(2(2(x1:S))) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 4#(5(2(x1:S))) -> 4#(x1:S) 4#(5(4(x1:S))) -> 4#(2(x1:S)) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = 1 [2](X) = X + 1 [3](X) = X [4](X) = 2.X + 1 [5](X) = X + 2 [0](X) = 0 [4#](X) = X Problem 1: SCC Processor: -> Pairs: 4#(5(4(x1:S))) -> 4#(2(x1:S)) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(5(4(x1:S))) -> 4#(2(x1:S)) ->->-> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 4#(5(4(x1:S))) -> 4#(2(x1:S)) -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) -> Usable rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [1](X) = 2 [2](X) = 2 [3](X) = 2 [4](X) = 2 [5](X) = 2.X [0](X) = 2 [4#](X) = X Problem 1: SCC Processor: -> Pairs: Empty -> Rules: 1(2(5(x1:S))) -> 1(2(2(1(0(1(x1:S)))))) 1(2(5(x1:S))) -> 1(0(5(0(5(4(x1:S)))))) 1(2(5(x1:S))) -> 2(0(1(3(1(0(x1:S)))))) 1(4(5(x1:S))) -> 1(2(4(0(2(1(x1:S)))))) 2(5(1(x1:S))) -> 2(2(2(1(2(3(x1:S)))))) 2(5(2(x1:S))) -> 4(0(2(2(3(3(x1:S)))))) 2(5(3(x1:S))) -> 2(0(4(1(3(3(x1:S)))))) 2(5(4(x1:S))) -> 2(0(5(1(0(1(x1:S)))))) 2(5(x1:S)) -> 1(3(3(0(1(0(x1:S)))))) 2(5(x1:S)) -> 2(2(0(5(0(1(x1:S)))))) 3(2(5(x1:S))) -> 3(2(0(1(0(5(x1:S)))))) 3(4(2(x1:S))) -> 3(4(0(2(2(2(x1:S)))))) 3(5(1(x1:S))) -> 2(1(4(1(0(1(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(2(3(4(x1:S)))))) 3(5(1(x1:S))) -> 0(4(2(0(0(5(x1:S)))))) 3(5(2(x1:S))) -> 2(3(3(2(1(2(x1:S)))))) 3(5(2(x1:S))) -> 2(0(2(2(3(0(x1:S)))))) 3(5(2(x1:S))) -> 0(4(3(2(2(2(x1:S)))))) 3(5(3(x1:S))) -> 2(3(4(0(4(2(x1:S)))))) 3(5(3(x1:S))) -> 0(2(4(3(3(0(x1:S)))))) 3(5(3(x1:S))) -> 0(5(4(3(3(0(x1:S)))))) 3(5(4(x1:S))) -> 0(2(0(5(0(0(x1:S)))))) 3(5(4(x1:S))) -> 0(5(0(0(1(2(x1:S)))))) 3(5(5(x1:S))) -> 0(5(4(1(0(5(x1:S)))))) 3(5(x1:S)) -> 1(3(2(0(0(1(x1:S)))))) 3(5(x1:S)) -> 3(2(0(5(3(0(x1:S)))))) 4(5(1(x1:S))) -> 2(1(0(5(3(3(x1:S)))))) 4(5(2(x1:S))) -> 0(5(1(0(0(4(x1:S)))))) 4(5(4(x1:S))) -> 2(2(1(0(4(2(x1:S)))))) 4(5(4(x1:S))) -> 3(2(0(3(2(0(x1:S)))))) 4(5(x1:S)) -> 2(2(1(3(2(1(x1:S)))))) 4(5(x1:S)) -> 3(2(0(5(0(0(x1:S)))))) 5(5(3(x1:S))) -> 5(1(0(1(2(2(x1:S)))))) 5(5(4(x1:S))) -> 5(1(0(4(2(2(x1:S)))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite.