YES Problem 1: (VAR v_NonEmpty:S x1:S) (RULES 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ) Problem 1: Dependency Pairs Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(0(1(x1:S))) -> 0#(0(2(2(3(1(x1:S)))))) 0#(0(1(x1:S))) -> 0#(2(1(3(x1:S)))) 0#(0(1(x1:S))) -> 0#(2(2(3(1(x1:S))))) 0#(0(1(x1:S))) -> 1#(3(x1:S)) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(1(2(0(x1:S)))) -> 1#(0(x1:S)) 0#(4(0(4(x1:S)))) -> 0#(0(3(2(4(4(x1:S)))))) 0#(4(0(4(x1:S)))) -> 0#(3(2(4(4(x1:S))))) 0#(4(0(4(x1:S)))) -> 4#(4(x1:S)) 0#(4(1(x1:S))) -> 0#(4(2(2(1(x1:S))))) 0#(4(1(x1:S))) -> 4#(2(2(1(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(5(4(1(1(x1:S))))) -> 5#(1(1(x1:S))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(2(0(1(x1:S)))) -> 1#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 0#(3(2(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(0(3(2(1(x1:S))))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(0(3(1(x1:S)))) -> 4#(2(3(1(1(0(x1:S)))))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(4(1(x1:S))) -> 4#(2(1(1(2(1(x1:S)))))) 1#(4(1(x1:S))) -> 4#(2(1(1(x1:S)))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 1#(2(3(3(1(4(x1:S)))))) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 1#(3(0(4(x1:S)))) -> 0#(3(2(1(3(x1:S))))) 1#(3(0(4(x1:S)))) -> 1#(3(x1:S)) 1#(3(0(4(x1:S)))) -> 4#(0(3(2(1(3(x1:S)))))) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(3(2(1(0(x1:S))))) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(0(3(1(x1:S))))) -> 4#(0(3(2(1(0(x1:S)))))) 4#(0(0(3(1(x1:S))))) -> 4#(3(2(0(0(1(x1:S)))))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 0#(2(4(2(1(5(x1:S)))))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 4#(2(1(5(x1:S)))) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(3(2(5(1(0(x1:S)))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 0#(2(3(5(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(3(2(1(5(5(x1:S)))))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(5(1(x1:S)))) -> 5#(2(5(3(1(0(x1:S)))))) 5#(0(5(1(x1:S)))) -> 5#(3(1(0(x1:S)))) 5#(0(5(x1:S))) -> 0#(3(2(5(3(5(x1:S)))))) 5#(0(5(x1:S))) -> 5#(3(5(x1:S))) 5#(0(3(1(x1:S)))) -> 0#(5(3(2(1(x1:S))))) 5#(0(3(1(x1:S)))) -> 0#(3(5(2(2(1(x1:S)))))) 5#(0(3(1(x1:S)))) -> 5#(2(2(1(x1:S)))) 5#(0(3(1(x1:S)))) -> 5#(3(2(1(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 4#(2(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 1#(3(x1:S)) 5#(4(1(x1:S))) -> 4#(2(5(1(3(x1:S))))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 1#(3(x1:S)) 5#(4(3(1(x1:S)))) -> 4#(2(1(3(x1:S)))) 5#(4(3(1(x1:S)))) -> 4#(2(1(x1:S))) 5#(4(3(1(x1:S)))) -> 4#(3(2(1(x1:S)))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(2(4(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(2(4(3(2(1(x1:S)))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 1#(4(2(5(x1:S)))) 5#(2(4(1(x1:S)))) -> 4#(2(5(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(0(1(x1:S))) -> 0#(0(2(2(3(1(x1:S)))))) 0#(0(1(x1:S))) -> 0#(2(1(3(x1:S)))) 0#(0(1(x1:S))) -> 0#(2(2(3(1(x1:S))))) 0#(0(1(x1:S))) -> 1#(3(x1:S)) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(1(2(0(x1:S)))) -> 1#(0(x1:S)) 0#(4(0(4(x1:S)))) -> 0#(0(3(2(4(4(x1:S)))))) 0#(4(0(4(x1:S)))) -> 0#(3(2(4(4(x1:S))))) 0#(4(0(4(x1:S)))) -> 4#(4(x1:S)) 0#(4(1(x1:S))) -> 0#(4(2(2(1(x1:S))))) 0#(4(1(x1:S))) -> 4#(2(2(1(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(5(4(1(1(x1:S))))) -> 5#(1(1(x1:S))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(2(0(1(x1:S)))) -> 1#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 0#(3(2(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(0(3(2(1(x1:S))))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(0(3(1(x1:S)))) -> 4#(2(3(1(1(0(x1:S)))))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(4(1(x1:S))) -> 4#(2(1(1(2(1(x1:S)))))) 1#(4(1(x1:S))) -> 4#(2(1(1(x1:S)))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 1#(2(3(3(1(4(x1:S)))))) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 1#(3(0(4(x1:S)))) -> 0#(3(2(1(3(x1:S))))) 1#(3(0(4(x1:S)))) -> 1#(3(x1:S)) 1#(3(0(4(x1:S)))) -> 4#(0(3(2(1(3(x1:S)))))) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(3(2(1(0(x1:S))))) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(0(3(1(x1:S))))) -> 4#(0(3(2(1(0(x1:S)))))) 4#(0(0(3(1(x1:S))))) -> 4#(3(2(0(0(1(x1:S)))))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 0#(2(4(2(1(5(x1:S)))))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 4#(2(1(5(x1:S)))) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(3(2(5(1(0(x1:S)))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 0#(2(3(5(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(3(2(1(5(5(x1:S)))))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(5(1(x1:S)))) -> 5#(2(5(3(1(0(x1:S)))))) 5#(0(5(1(x1:S)))) -> 5#(3(1(0(x1:S)))) 5#(0(5(x1:S))) -> 0#(3(2(5(3(5(x1:S)))))) 5#(0(5(x1:S))) -> 5#(3(5(x1:S))) 5#(0(3(1(x1:S)))) -> 0#(5(3(2(1(x1:S))))) 5#(0(3(1(x1:S)))) -> 0#(3(5(2(2(1(x1:S)))))) 5#(0(3(1(x1:S)))) -> 5#(2(2(1(x1:S)))) 5#(0(3(1(x1:S)))) -> 5#(3(2(1(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 4#(2(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 1#(3(x1:S)) 5#(4(1(x1:S))) -> 4#(2(5(1(3(x1:S))))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 1#(3(x1:S)) 5#(4(3(1(x1:S)))) -> 4#(2(1(3(x1:S)))) 5#(4(3(1(x1:S)))) -> 4#(2(1(x1:S))) 5#(4(3(1(x1:S)))) -> 4#(3(2(1(x1:S)))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(2(4(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(2(4(3(2(1(x1:S)))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 1#(4(2(5(x1:S)))) 5#(2(4(1(x1:S)))) -> 4#(2(5(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(3(0(4(x1:S)))) -> 1#(3(x1:S)) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(0(1(x1:S))) -> 0#(2(1(3(x1:S)))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(1(2(0(x1:S)))) -> 1#(0(x1:S)) 0#(4(0(4(x1:S)))) -> 4#(4(x1:S)) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(5(4(1(1(x1:S))))) -> 5#(1(1(x1:S))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) The problem is decomposed in 2 subproblems. Problem 1.1: Reduction Pair Processor: -> Pairs: 1#(3(0(4(x1:S)))) -> 1#(3(x1:S)) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: Empty ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [4](X) = X + 2 [3](X) = 2.X [1#](X) = 2.X Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(0(1(x1:S))) -> 0#(2(1(3(x1:S)))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(1(2(0(x1:S)))) -> 1#(0(x1:S)) 0#(4(0(4(x1:S)))) -> 4#(4(x1:S)) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(5(4(1(1(x1:S))))) -> 5#(1(1(x1:S))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 1 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X + 1 [4#](X) = X + 1 [5#](X) = X + 1 Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(1(2(0(x1:S)))) -> 1#(0(x1:S)) 0#(4(0(4(x1:S)))) -> 4#(4(x1:S)) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(5(4(1(1(x1:S))))) -> 5#(1(1(x1:S))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(1(2(0(x1:S)))) -> 1#(0(x1:S)) 0#(4(0(4(x1:S)))) -> 4#(4(x1:S)) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(5(4(1(1(x1:S))))) -> 5#(1(1(x1:S))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(1(2(0(x1:S)))) -> 1#(0(x1:S)) 0#(4(0(4(x1:S)))) -> 4#(4(x1:S)) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(5(4(1(1(x1:S))))) -> 5#(1(1(x1:S))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(4(0(4(x1:S)))) -> 4#(4(x1:S)) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(5(4(1(1(x1:S))))) -> 5#(1(1(x1:S))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(4(0(4(x1:S)))) -> 4#(4(x1:S)) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(5(4(1(1(x1:S))))) -> 5#(1(1(x1:S))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(4(0(4(x1:S)))) -> 4#(4(x1:S)) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(5(4(1(1(x1:S))))) -> 5#(1(1(x1:S))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(5(4(1(1(x1:S))))) -> 5#(1(1(x1:S))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(5(4(1(1(x1:S))))) -> 5#(1(1(x1:S))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(5(4(1(1(x1:S))))) -> 5#(1(1(x1:S))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(2(0(1(x1:S)))) -> 0#(2(1(3(x1:S)))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 1 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X + 1 [4#](X) = X + 1 [5#](X) = X + 1 Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 0#(3(x1:S)) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(2(x1:S))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(1(x1:S)) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(1(x1:S))) -> 1#(2(x1:S)) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(1(x1:S)) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 0#(x1:S) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(0(3(1(x1:S))))) -> 1#(0(x1:S)) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 1#(5(x1:S)) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(0(5(1(x1:S)))) -> 5#(x1:S) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 1#(x1:S) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 1 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X + 1 [4#](X) = X + 1 [5#](X) = X + 1 Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 0#(x1:S) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 1#(0(x1:S)) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(0(1(x1:S)))) -> 5#(1(0(x1:S))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 1#(5(5(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(1(x1:S)) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(5(x1:S)) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(1(x1:S))) -> 5#(x1:S) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 0#(2(1(5(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 1#(5(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X [4#](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(5(0(4(x1:S)))) -> 5#(4(2(x1:S))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 1 [1](X) = X [4](X) = X [5](X) = X [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = X + 1 [4#](X) = X + 1 [5#](X) = X + 1 Problem 1.2: SCC Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(0(1(x1:S))) -> 0#(0(2(1(3(x1:S))))) 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X [1](X) = 2.X + 2 [4](X) = 2.X [5](X) = X + 1 [2](X) = 1/2.X [3](X) = X [0#](X) = X [1#](X) = X + 1 [4#](X) = 2.X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(1(2(0(x1:S)))) -> 0#(2(1(0(x1:S)))) 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 1 [4](X) = 2.X + 1/2 [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [0#](X) = 1/2.X [1#](X) = 1/2.X [4#](X) = 1/2.X [5#](X) = 1/2.X + 1 Problem 1.2: SCC Processor: -> Pairs: 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(5(4(1(1(x1:S))))) -> 0#(2(5(1(1(x1:S))))) 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 1 [4](X) = 2.X + 1 [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [0#](X) = X + 1/2 [1#](X) = 1/2.X + 1 [4#](X) = X + 1 [5#](X) = X + 1/2 Problem 1.2: SCC Processor: -> Pairs: 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(5(4(1(1(x1:S))))) -> 4#(0(2(5(1(1(x1:S)))))) 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X [1](X) = 2.X + 2 [4](X) = 2.X [5](X) = X + 1 [2](X) = 1/2.X [3](X) = X [0#](X) = X + 1 [1#](X) = 2.X + 1 [4#](X) = 2.X + 1 [5#](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 0#(0(3(x1:S))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 1 [4](X) = 2.X + 1/2 [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [0#](X) = X + 1/2 [1#](X) = X [4#](X) = X [5#](X) = 1/2.X + 2 Problem 1.2: SCC Processor: -> Pairs: 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 0#(3(4(0(x1:S)))) -> 4#(0(0(3(x1:S)))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 2 [4](X) = 2.X + 1 [5](X) = X + 1/2 [2](X) = 1/2.X [3](X) = X [0#](X) = 1/2.X + 1/2 [1#](X) = 1/2.X + 1 [4#](X) = 1/2.X + 1/2 [5#](X) = 1/2.X Problem 1.2: SCC Processor: -> Pairs: 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) 1#(0(1(4(x1:S)))) -> 0#(x1:S) 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(1(x1:S))) -> 0#(2(1(1(2(x1:S))))) 1#(0(1(x1:S))) -> 0#(2(1(1(x1:S)))) 1#(0(3(1(x1:S)))) -> 0#(x1:S) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(0(3(1(x1:S))))) -> 0#(0(1(x1:S))) 4#(0(1(x1:S))) -> 0#(2(1(x1:S))) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 0#(2(1(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 0#(2(1(x1:S))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 0#(x1:S) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 0#(0(2(1(5(x1:S))))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 0#(2(1(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 0#(2(5(4(2(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->->Cycle: ->->-> Pairs: 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) The problem is decomposed in 2 subproblems. Problem 1.2.1: Reduction Pair Processor: -> Pairs: 0#(2(0(1(x1:S)))) -> 0#(0(2(1(3(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 1/2.X [1](X) = 2 [4](X) = 2 [5](X) = 2.X + 1/2 [2](X) = 1/2 [3](X) = 0 [0#](X) = 1/2.X Problem 1.2.1: SCC Processor: -> Pairs: Empty -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2.2: Reduction Pair Processor: -> Pairs: 1#(0(1(4(x1:S)))) -> 1#(1(4(0(x1:S)))) 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 2 [4](X) = 2.X + 1 [5](X) = X + 1/2 [2](X) = 1/2.X [3](X) = X [1#](X) = 2.X + 1/2 [4#](X) = 2.X + 2 [5#](X) = 2.X Problem 1.2.2: SCC Processor: -> Pairs: 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2: Reduction Pair Processor: -> Pairs: 1#(0(1(4(x1:S)))) -> 1#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 2 [4](X) = 2.X + 1 [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [1#](X) = 1/2.X + 1/2 [4#](X) = X + 1 [5#](X) = X Problem 1.2.2: SCC Processor: -> Pairs: 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2: Reduction Pair Processor: -> Pairs: 1#(0(1(4(x1:S)))) -> 4#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 1 [4](X) = 2.X + 1/2 [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [1#](X) = X + 2 [4#](X) = X + 2 [5#](X) = 2.X + 1 Problem 1.2.2: SCC Processor: -> Pairs: 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2: Reduction Pair Processor: -> Pairs: 1#(0(3(1(x1:S)))) -> 1#(0(x1:S)) 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 1 [4](X) = 2.X + 1/2 [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [1#](X) = 2.X + 1 [4#](X) = 2.X + 2 [5#](X) = 2.X Problem 1.2.2: SCC Processor: -> Pairs: 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2: Reduction Pair Processor: -> Pairs: 1#(0(3(1(x1:S)))) -> 1#(1(0(x1:S))) 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 1 [4](X) = 2.X + 1/2 [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [1#](X) = X [4#](X) = X + 1/2 [5#](X) = 1/2.X + 2 Problem 1.2.2: SCC Processor: -> Pairs: 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2: Reduction Pair Processor: -> Pairs: 1#(4(1(5(x1:S)))) -> 1#(4(x1:S)) 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X [1](X) = 2.X + 2 [4](X) = 2.X [5](X) = X + 1 [2](X) = 1/2.X [3](X) = X [1#](X) = 1/2.X + 2 [4#](X) = X + 1/2 [5#](X) = 1/2.X + 1/2 Problem 1.2.2: SCC Processor: -> Pairs: 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2: Reduction Pair Processor: -> Pairs: 1#(4(1(5(x1:S)))) -> 1#(2(5(1(4(x1:S))))) 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X [1](X) = 2.X + 2 [4](X) = 2.X [5](X) = 2.X + 1 [2](X) = 1/2.X [3](X) = X [1#](X) = X [4#](X) = X [5#](X) = X + 1/2 Problem 1.2.2: SCC Processor: -> Pairs: 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2: Reduction Pair Processor: -> Pairs: 1#(4(1(5(x1:S)))) -> 4#(x1:S) 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = X + 1/2 [2](X) = 1/2.X [3](X) = X [1#](X) = X [4#](X) = X [5#](X) = 1/2.X Problem 1.2.2: SCC Processor: -> Pairs: 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2: Reduction Pair Processor: -> Pairs: 1#(4(1(5(x1:S)))) -> 5#(1(4(x1:S))) 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 1 [4](X) = 2.X + 1 [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [1#](X) = X + 2 [4#](X) = X + 2 [5#](X) = 2.X Problem 1.2.2: SCC Processor: -> Pairs: 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(0(5(1(x1:S)))) -> 1#(0(x1:S)) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 1#(5(4(2(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 1#(x1:S) 5#(4(1(5(x1:S)))) -> 4#(5(5(2(1(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 4#(0(2(1(x1:S)))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->->Cycle: ->->-> Pairs: 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) The problem is decomposed in 2 subproblems. Problem 1.2.2.1: Reduction Pair Processor: -> Pairs: 1#(4(1(x1:S))) -> 1#(1(2(1(x1:S)))) 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 2 [4](X) = 2.X + 1 [5](X) = 2.X + 1 [2](X) = 1/2.X [3](X) = X [1#](X) = X [4#](X) = 2.X Problem 1.2.2.1: SCC Processor: -> Pairs: 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.1: Reduction Pair Processor: -> Pairs: 1#(4(1(x1:S))) -> 1#(1(x1:S)) 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [1#](X) = 1/2.X + 1/2 [4#](X) = 1/2.X + 1/2 Problem 1.2.2.1: SCC Processor: -> Pairs: 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.1: Reduction Pair Processor: -> Pairs: 1#(4(1(x1:S))) -> 1#(2(1(x1:S))) 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X [1](X) = 2.X + 1 [4](X) = 2.X [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [1#](X) = 1/2.X + 1 [4#](X) = 1/2.X + 1/2 Problem 1.2.2.1: SCC Processor: -> Pairs: 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.1: Reduction Pair Processor: -> Pairs: 1#(2(4(3(1(x1:S))))) -> 1#(4(x1:S)) 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 1 [4](X) = X [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [1#](X) = X [4#](X) = 1/2.X Problem 1.2.2.1: SCC Processor: -> Pairs: 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.1: Reduction Pair Processor: -> Pairs: 1#(2(4(3(1(x1:S))))) -> 4#(x1:S) 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 0 [1](X) = 2 [4](X) = X [5](X) = 2 [2](X) = X [3](X) = X [1#](X) = X + 2 [4#](X) = 2 Problem 1.2.2.1: SCC Processor: -> Pairs: 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 1#(4(0(0(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.1: Reduction Pair Processor: -> Pairs: 4#(0(1(x1:S))) -> 4#(0(2(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 1 [4](X) = 2.X [5](X) = 2.X + 2 [2](X) = 0 [3](X) = 2.X [4#](X) = 2.X Problem 1.2.2.1: SCC Processor: -> Pairs: 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.1: Reduction Pair Processor: -> Pairs: 4#(1(0(0(x1:S)))) -> 4#(0(0(x1:S))) 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 0 [1](X) = 2 [4](X) = 0 [5](X) = 2 [2](X) = 0 [3](X) = 0 [4#](X) = X Problem 1.2.2.1: SCC Processor: -> Pairs: 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.1: Reduction Pair Processor: -> Pairs: 4#(1(0(4(x1:S)))) -> 4#(0(2(1(x1:S)))) 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 0 [1](X) = 2 [4](X) = 0 [5](X) = 0 [2](X) = 0 [3](X) = 2.X [4#](X) = 2.X Problem 1.2.2.1: SCC Processor: -> Pairs: 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.1: Reduction Pair Processor: -> Pairs: 4#(1(0(4(x1:S)))) -> 4#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X [1](X) = 2.X + 2 [4](X) = 2.X [5](X) = X + 2 [2](X) = 0 [3](X) = 0 [4#](X) = 2.X Problem 1.2.2.1: SCC Processor: -> Pairs: Empty -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2.2.2: Reduction Pair Processor: -> Pairs: 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(2(1(x1:S))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X [1](X) = X [4](X) = X [5](X) = 2.X + 2 [2](X) = X [3](X) = 0 [5#](X) = 2.X Problem 1.2.2.2: SCC Processor: -> Pairs: 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.2: Reduction Pair Processor: -> Pairs: 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(5(0(4(x1:S)))) -> 5#(0(2(5(4(2(x1:S)))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 0 [1](X) = X [4](X) = X [5](X) = 2 [2](X) = X [3](X) = 0 [5#](X) = 2.X Problem 1.2.2.2: SCC Processor: -> Pairs: 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.2: Reduction Pair Processor: -> Pairs: 5#(0(1(x1:S))) -> 5#(0(2(1(x1:S)))) 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = X + 1 [4](X) = 2.X + 1 [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [5#](X) = 1/2.X Problem 1.2.2.2: SCC Processor: -> Pairs: 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.2: Reduction Pair Processor: -> Pairs: 5#(1(0(0(5(x1:S))))) -> 5#(0(0(2(1(5(x1:S)))))) 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = X + 1 [4](X) = 2.X + 1 [5](X) = X + 1 [2](X) = 1/2.X [3](X) = X [5#](X) = 2.X Problem 1.2.2.2: SCC Processor: -> Pairs: 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.2: Reduction Pair Processor: -> Pairs: 5#(4(1(5(x1:S)))) -> 5#(4(2(5(x1:S)))) 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X [1](X) = X + 2 [4](X) = 2.X [5](X) = X + 2 [2](X) = 1/2.X [3](X) = X [5#](X) = 1/2.X Problem 1.2.2.2: SCC Processor: -> Pairs: 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.2: Reduction Pair Processor: -> Pairs: 5#(4(1(5(x1:S)))) -> 5#(5(2(1(x1:S)))) 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = X + 1/2 [4](X) = 2.X + 1/2 [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [5#](X) = 1/2.X Problem 1.2.2.2: SCC Processor: -> Pairs: 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.2: Reduction Pair Processor: -> Pairs: 5#(4(1(x1:S))) -> 5#(1(3(x1:S))) 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X [1](X) = 2.X + 1 [4](X) = 2.X [5](X) = 2.X + 1/2 [2](X) = 1/2.X [3](X) = X [5#](X) = X Problem 1.2.2.2: SCC Processor: -> Pairs: 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.2: Reduction Pair Processor: -> Pairs: 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(4(3(1(x1:S)))) -> 5#(4(2(1(3(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X [1](X) = X + 1 [4](X) = 2.X [5](X) = 1/2.X [2](X) = 1/2.X [3](X) = X [5#](X) = 1/2.X Problem 1.2.2.2: SCC Processor: -> Pairs: 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.2: Reduction Pair Processor: -> Pairs: 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) 5#(2(4(1(x1:S)))) -> 5#(x1:S) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = X + 1/2 [4](X) = 2.X + 1/2 [5](X) = X + 1/2 [2](X) = 1/2.X [3](X) = X [5#](X) = 2.X Problem 1.2.2.2: SCC Processor: -> Pairs: 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) ->->-> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) Problem 1.2.2.2: Reduction Pair Processor: -> Pairs: 5#(4(2(0(1(x1:S))))) -> 5#(4(0(2(1(x1:S))))) -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) -> Usable rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 1/2.X [1](X) = 2 [4](X) = 2.X + 1 [5](X) = 1/2.X + 1/2 [2](X) = 1/2 [3](X) = 2.X [5#](X) = X Problem 1.2.2.2: SCC Processor: -> Pairs: Empty -> Rules: 0(0(1(x1:S))) -> 0(0(2(1(3(x1:S))))) 0(0(1(x1:S))) -> 0(0(2(2(3(1(x1:S)))))) 0(0(4(5(x1:S)))) -> 0(5(0(2(4(2(x1:S)))))) 0(0(3(3(1(x1:S))))) -> 0(3(0(1(2(3(x1:S)))))) 0(1(2(0(x1:S)))) -> 2(0(2(1(0(x1:S))))) 0(4(0(4(x1:S)))) -> 0(0(3(2(4(4(x1:S)))))) 0(4(1(x1:S))) -> 0(4(2(2(1(x1:S))))) 0(4(1(x1:S))) -> 0(2(3(1(4(2(x1:S)))))) 0(5(4(1(1(x1:S))))) -> 4(0(2(5(1(1(x1:S)))))) 0(2(0(1(x1:S)))) -> 0(0(2(1(3(x1:S))))) 0(2(4(3(1(x1:S))))) -> 0(2(3(1(4(2(x1:S)))))) 0(3(4(0(x1:S)))) -> 3(2(4(0(0(3(x1:S)))))) 1(0(1(4(x1:S)))) -> 2(3(1(1(4(0(x1:S)))))) 1(0(1(x1:S))) -> 0(2(1(1(2(x1:S))))) 1(0(1(x1:S))) -> 0(2(1(1(x1:S)))) 1(0(1(x1:S))) -> 1(0(3(2(1(x1:S))))) 1(0(3(1(x1:S)))) -> 4(2(3(1(1(0(x1:S)))))) 1(4(0(x1:S))) -> 1(0(3(2(4(2(x1:S)))))) 1(4(1(5(x1:S)))) -> 2(1(2(5(1(4(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(2(1(x1:S)))))) 1(4(1(x1:S))) -> 4(2(1(1(x1:S)))) 1(2(0(4(x1:S)))) -> 1(4(2(0(3(2(x1:S)))))) 1(2(4(3(1(x1:S))))) -> 1(2(3(3(1(4(x1:S)))))) 1(3(0(4(x1:S)))) -> 4(0(3(2(1(3(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 4(0(0(3(1(x1:S))))) -> 4(3(2(0(0(1(x1:S)))))) 4(0(1(x1:S))) -> 4(0(2(1(x1:S)))) 4(0(5(1(x1:S)))) -> 0(2(4(2(1(5(x1:S)))))) 4(1(0(0(x1:S)))) -> 3(2(1(4(0(0(x1:S)))))) 4(1(0(4(x1:S)))) -> 4(4(0(2(1(x1:S))))) 5(0(0(1(x1:S)))) -> 0(3(2(5(1(0(x1:S)))))) 5(0(1(x1:S))) -> 0(2(1(5(3(x1:S))))) 5(0(1(x1:S))) -> 0(3(2(1(5(5(x1:S)))))) 5(0(1(x1:S))) -> 5(0(2(1(x1:S)))) 5(0(1(x1:S))) -> 3(0(2(3(5(1(x1:S)))))) 5(0(5(1(x1:S)))) -> 5(2(5(3(1(0(x1:S)))))) 5(0(5(x1:S))) -> 0(3(2(5(3(5(x1:S)))))) 5(0(3(1(x1:S)))) -> 0(5(3(2(1(x1:S))))) 5(0(3(1(x1:S)))) -> 0(3(5(2(2(1(x1:S)))))) 5(1(0(0(5(x1:S))))) -> 5(0(0(2(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(5(5(2(1(x1:S))))) 5(4(1(5(x1:S)))) -> 2(1(5(4(2(5(x1:S)))))) 5(4(1(x1:S))) -> 4(5(2(1(3(3(x1:S)))))) 5(4(1(x1:S))) -> 2(4(2(5(1(3(x1:S)))))) 5(4(2(0(1(x1:S))))) -> 2(5(4(0(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 5(4(2(1(3(x1:S))))) 5(4(3(1(x1:S)))) -> 5(2(4(3(2(1(x1:S)))))) 5(4(3(1(x1:S)))) -> 3(5(2(4(2(1(x1:S)))))) 5(5(0(4(x1:S)))) -> 5(0(2(5(4(2(x1:S)))))) 5(2(4(1(x1:S)))) -> 2(1(4(2(5(x1:S))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite.