rm: cannot remove ‘/tmp/muTerm5117023052084420925.in.dec’: No such file or directory rm: cannot remove ‘/tmp/muTerm5117023052084420925.in.form’: No such file or directory rm: cannot remove ‘/tmp/muTerm5726603361159126505.in.dec’: No such file or directory rm: cannot remove ‘/tmp/muTerm5726603361159126505.in.form’: No such file or directory rm: cannot remove ‘/tmp/muTerm9398195822001100545.in.dec’: No such file or directory rm: cannot remove ‘/tmp/muTerm9398195822001100545.in.form’: No such file or directory rm: cannot remove ‘/tmp/muTerm6105154341585990364.in.dec’: No such file or directory rm: cannot remove ‘/tmp/muTerm6105154341585990364.in.form’: No such file or directory rm: cannot remove ‘/tmp/muTerm7523927541474612399.in.dec’: No such file or directory rm: cannot remove ‘/tmp/muTerm7523927541474612399.in.form’: No such file or directory YES Problem 1: (VAR v_NonEmpty:S x1:S) (RULES 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ) Problem 1: Dependency Pairs Processor: -> Pairs: 0#(1(0(0(x1:S)))) -> 0#(0(1(x1:S))) 0#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 0#(1(0(0(x1:S)))) -> 0#(2(0(0(1(x1:S))))) 0#(1(0(0(x1:S)))) -> 1#(x1:S) 1#(0(0(1(x1:S)))) -> 0#(3(0(1(x1:S)))) 1#(0(0(1(x1:S)))) -> 1#(0(3(0(1(x1:S))))) 1#(0(0(4(x1:S)))) -> 0#(2(0(x1:S))) 1#(0(0(4(x1:S)))) -> 0#(3(2(0(x1:S)))) 1#(0(0(4(x1:S)))) -> 0#(x1:S) 1#(0(0(4(x1:S)))) -> 1#(4(0(2(0(x1:S))))) 1#(0(0(4(x1:S)))) -> 1#(4(0(3(2(0(x1:S)))))) 1#(0(0(4(x1:S)))) -> 4#(0(2(0(x1:S)))) 1#(0(0(4(x1:S)))) -> 4#(0(3(2(0(x1:S))))) 1#(0(1(0(x1:S)))) -> 0#(0(3(1(x1:S)))) 1#(0(1(0(x1:S)))) -> 0#(3(1(x1:S))) 1#(0(1(0(x1:S)))) -> 1#(3(0(0(3(1(x1:S)))))) 1#(0(1(0(x1:S)))) -> 1#(x1:S) 1#(0(1(4(5(x1:S))))) -> 0#(x1:S) 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 1#(3(1(5(4(0(x1:S)))))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 0#(3(1(2(4(x1:S))))) 1#(0(1(4(x1:S)))) -> 1#(0(3(1(2(4(x1:S)))))) 1#(0(1(4(x1:S)))) -> 1#(2(4(x1:S))) 1#(0(1(5(x1:S)))) -> 0#(2(1(5(x1:S)))) 1#(0(1(5(x1:S)))) -> 1#(0(2(1(5(x1:S))))) 1#(0(4(0(4(x1:S))))) -> 0#(4(3(4(0(x1:S))))) 1#(0(4(0(4(x1:S))))) -> 0#(x1:S) 1#(0(4(0(4(x1:S))))) -> 1#(0(4(3(4(0(x1:S)))))) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(0(4(0(4(x1:S))))) -> 4#(3(4(0(x1:S)))) 1#(0(4(1(x1:S)))) -> 0#(3(4(1(x1:S)))) 1#(0(4(1(x1:S)))) -> 1#(0(3(4(1(x1:S))))) 1#(0(5(0(x1:S)))) -> 0#(0(x1:S)) 1#(0(5(0(x1:S)))) -> 1#(2(5(2(0(0(x1:S)))))) 1#(0(5(0(x1:S)))) -> 5#(2(0(0(x1:S)))) 1#(1(1(0(0(x1:S))))) -> 0#(0(3(1(x1:S)))) 1#(1(1(0(0(x1:S))))) -> 0#(3(1(x1:S))) 1#(1(1(0(0(x1:S))))) -> 1#(0(0(3(1(x1:S))))) 1#(1(1(0(0(x1:S))))) -> 1#(1(0(0(3(1(x1:S)))))) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 0#(5(x1:S)) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 0#(2(1(x1:S))) 1#(4(1(0(x1:S)))) -> 0#(3(1(x1:S))) 1#(4(1(0(x1:S)))) -> 1#(4(0(2(1(x1:S))))) 1#(4(1(0(x1:S)))) -> 1#(4(0(3(1(x1:S))))) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(1(0(x1:S)))) -> 4#(0(2(1(x1:S)))) 1#(4(1(0(x1:S)))) -> 4#(0(3(1(x1:S)))) 1#(4(5(0(0(x1:S))))) -> 0#(5(0(2(1(x1:S))))) 1#(4(5(0(0(x1:S))))) -> 0#(2(1(x1:S))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 4#(0(5(0(2(1(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 5#(0(2(1(x1:S)))) 1#(4(5(0(x1:S)))) -> 0#(2(1(2(5(x1:S))))) 1#(4(5(0(x1:S)))) -> 1#(2(5(x1:S))) 1#(4(5(0(x1:S)))) -> 4#(0(2(1(2(5(x1:S)))))) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 0#(3(1(2(5(4(x1:S)))))) 1#(4(5(2(0(x1:S))))) -> 1#(2(5(4(x1:S)))) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 0#(1(3(1(3(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 0#(3(1(3(x1:S)))) 1#(5(0(1(x1:S)))) -> 1#(5(0(3(1(3(x1:S)))))) 1#(5(0(1(x1:S)))) -> 1#(3(1(3(5(x1:S))))) 1#(5(0(1(x1:S)))) -> 1#(3(5(x1:S))) 1#(5(0(1(x1:S)))) -> 1#(3(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(0(3(1(3(x1:S))))) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 0#(5(1(5(x1:S)))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 1#(2(0(4(x1:S)))) -> 0#(x1:S) 1#(2(0(4(x1:S)))) -> 1#(3(4(2(0(x1:S))))) 1#(2(0(4(x1:S)))) -> 4#(2(0(x1:S))) 1#(2(0(5(0(x1:S))))) -> 0#(2(0(x1:S))) 1#(2(0(5(0(x1:S))))) -> 1#(0(2(0(x1:S)))) 1#(2(0(5(0(x1:S))))) -> 5#(2(1(0(2(0(x1:S)))))) 1#(3(0(4(x1:S)))) -> 0#(x1:S) 1#(3(0(4(x1:S)))) -> 1#(3(4(2(0(x1:S))))) 1#(3(0(4(x1:S)))) -> 4#(2(0(x1:S))) 4#(1(0(0(1(x1:S))))) -> 0#(4(0(3(1(x1:S))))) 4#(1(0(0(1(x1:S))))) -> 0#(3(1(x1:S))) 4#(1(0(0(1(x1:S))))) -> 1#(0(4(0(3(1(x1:S)))))) 4#(1(0(0(1(x1:S))))) -> 4#(0(3(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 4#(1(0(0(x1:S)))) -> 0#(2(4(2(0(1(x1:S)))))) 4#(1(0(0(x1:S)))) -> 0#(3(4(3(0(1(x1:S)))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(0(x1:S)))) -> 4#(2(0(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(3(0(1(x1:S)))) 4#(1(0(5(x1:S)))) -> 0#(2(1(5(x1:S)))) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(0(5(x1:S)))) -> 4#(0(2(1(5(x1:S))))) 4#(1(5(3(1(x1:S))))) -> 1#(5(4(3(1(x1:S))))) 4#(1(5(3(1(x1:S))))) -> 4#(3(1(x1:S))) 4#(1(5(3(1(x1:S))))) -> 5#(4(3(1(x1:S)))) 4#(1(2(0(0(x1:S))))) -> 0#(0(2(1(x1:S)))) 4#(1(2(0(0(x1:S))))) -> 0#(2(1(x1:S))) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(0(x1:S))))) -> 4#(2(0(0(2(1(x1:S)))))) 4#(1(2(0(x1:S)))) -> 0#(2(1(3(4(x1:S))))) 4#(1(2(0(x1:S)))) -> 1#(3(4(x1:S))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 0#(1(5(x1:S))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 1#(3(4(4(0(1(x1:S)))))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(4(0(5(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(5(x1:S)) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 0#(0(5(3(1(x1:S))))) 4#(5(0(1(0(x1:S))))) -> 0#(5(3(1(x1:S)))) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(0(1(0(x1:S))))) -> 4#(0(0(5(3(1(x1:S)))))) 4#(5(0(1(0(x1:S))))) -> 5#(3(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 0#(5(x1:S)) 4#(5(1(1(0(x1:S))))) -> 0#(2(1(5(1(x1:S))))) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 4#(0(2(1(5(1(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 4#(5(1(5(1(x1:S))))) -> 1#(5(5(3(1(x1:S))))) 4#(5(1(5(1(x1:S))))) -> 4#(1(5(5(3(1(x1:S)))))) 4#(5(1(5(1(x1:S))))) -> 5#(5(3(1(x1:S)))) 4#(5(1(5(1(x1:S))))) -> 5#(3(1(x1:S))) 5#(0(0(4(1(x1:S))))) -> 0#(1(x1:S)) 5#(0(0(4(1(x1:S))))) -> 0#(3(0(1(x1:S)))) 5#(0(0(4(1(x1:S))))) -> 4#(0(3(0(1(x1:S))))) 5#(0(0(4(1(x1:S))))) -> 5#(4(0(3(0(1(x1:S)))))) 5#(0(0(4(5(x1:S))))) -> 0#(0(3(5(5(x1:S))))) 5#(0(0(4(5(x1:S))))) -> 0#(5(5(x1:S))) 5#(0(0(4(5(x1:S))))) -> 0#(3(4(0(5(5(x1:S)))))) 5#(0(0(4(5(x1:S))))) -> 0#(3(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 4#(0(0(3(5(5(x1:S)))))) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(0(4(x1:S)))) -> 0#(2(0(x1:S))) 5#(0(0(4(x1:S)))) -> 0#(x1:S) 5#(0(0(4(x1:S)))) -> 4#(0(2(0(x1:S)))) 5#(0(0(4(x1:S)))) -> 5#(4(0(2(0(x1:S))))) 5#(0(1(4(x1:S)))) -> 0#(x1:S) 5#(0(1(4(x1:S)))) -> 1#(5(4(3(0(x1:S))))) 5#(0(1(4(x1:S)))) -> 4#(3(0(x1:S))) 5#(0(1(4(x1:S)))) -> 5#(4(3(0(x1:S)))) 5#(0(4(1(0(x1:S))))) -> 0#(4(0(3(1(x1:S))))) 5#(0(4(1(0(x1:S))))) -> 0#(3(1(x1:S))) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(0(4(1(0(x1:S))))) -> 4#(0(3(1(x1:S)))) 5#(0(4(1(0(x1:S))))) -> 5#(0(4(0(3(1(x1:S)))))) 5#(1(5(0(1(x1:S))))) -> 0#(3(5(1(1(x1:S))))) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(0(3(5(1(1(x1:S)))))) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 0#(1(5(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 1#(2(4(0(1(5(x1:S)))))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 4#(3(1(5(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 1#(3(4(0(x1:S)))) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) 5#(5(2(0(4(x1:S))))) -> 0#(x1:S) 5#(5(2(0(4(x1:S))))) -> 4#(2(0(x1:S))) 5#(5(2(0(4(x1:S))))) -> 5#(5(3(4(2(0(x1:S)))))) 5#(5(2(0(4(x1:S))))) -> 5#(3(4(2(0(x1:S))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: SCC Processor: -> Pairs: 0#(1(0(0(x1:S)))) -> 0#(0(1(x1:S))) 0#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 0#(1(0(0(x1:S)))) -> 0#(2(0(0(1(x1:S))))) 0#(1(0(0(x1:S)))) -> 1#(x1:S) 1#(0(0(1(x1:S)))) -> 0#(3(0(1(x1:S)))) 1#(0(0(1(x1:S)))) -> 1#(0(3(0(1(x1:S))))) 1#(0(0(4(x1:S)))) -> 0#(2(0(x1:S))) 1#(0(0(4(x1:S)))) -> 0#(3(2(0(x1:S)))) 1#(0(0(4(x1:S)))) -> 0#(x1:S) 1#(0(0(4(x1:S)))) -> 1#(4(0(2(0(x1:S))))) 1#(0(0(4(x1:S)))) -> 1#(4(0(3(2(0(x1:S)))))) 1#(0(0(4(x1:S)))) -> 4#(0(2(0(x1:S)))) 1#(0(0(4(x1:S)))) -> 4#(0(3(2(0(x1:S))))) 1#(0(1(0(x1:S)))) -> 0#(0(3(1(x1:S)))) 1#(0(1(0(x1:S)))) -> 0#(3(1(x1:S))) 1#(0(1(0(x1:S)))) -> 1#(3(0(0(3(1(x1:S)))))) 1#(0(1(0(x1:S)))) -> 1#(x1:S) 1#(0(1(4(5(x1:S))))) -> 0#(x1:S) 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 1#(3(1(5(4(0(x1:S)))))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 0#(3(1(2(4(x1:S))))) 1#(0(1(4(x1:S)))) -> 1#(0(3(1(2(4(x1:S)))))) 1#(0(1(4(x1:S)))) -> 1#(2(4(x1:S))) 1#(0(1(5(x1:S)))) -> 0#(2(1(5(x1:S)))) 1#(0(1(5(x1:S)))) -> 1#(0(2(1(5(x1:S))))) 1#(0(4(0(4(x1:S))))) -> 0#(4(3(4(0(x1:S))))) 1#(0(4(0(4(x1:S))))) -> 0#(x1:S) 1#(0(4(0(4(x1:S))))) -> 1#(0(4(3(4(0(x1:S)))))) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(0(4(0(4(x1:S))))) -> 4#(3(4(0(x1:S)))) 1#(0(4(1(x1:S)))) -> 0#(3(4(1(x1:S)))) 1#(0(4(1(x1:S)))) -> 1#(0(3(4(1(x1:S))))) 1#(0(5(0(x1:S)))) -> 0#(0(x1:S)) 1#(0(5(0(x1:S)))) -> 1#(2(5(2(0(0(x1:S)))))) 1#(0(5(0(x1:S)))) -> 5#(2(0(0(x1:S)))) 1#(1(1(0(0(x1:S))))) -> 0#(0(3(1(x1:S)))) 1#(1(1(0(0(x1:S))))) -> 0#(3(1(x1:S))) 1#(1(1(0(0(x1:S))))) -> 1#(0(0(3(1(x1:S))))) 1#(1(1(0(0(x1:S))))) -> 1#(1(0(0(3(1(x1:S)))))) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 0#(5(x1:S)) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 0#(2(1(x1:S))) 1#(4(1(0(x1:S)))) -> 0#(3(1(x1:S))) 1#(4(1(0(x1:S)))) -> 1#(4(0(2(1(x1:S))))) 1#(4(1(0(x1:S)))) -> 1#(4(0(3(1(x1:S))))) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(1(0(x1:S)))) -> 4#(0(2(1(x1:S)))) 1#(4(1(0(x1:S)))) -> 4#(0(3(1(x1:S)))) 1#(4(5(0(0(x1:S))))) -> 0#(5(0(2(1(x1:S))))) 1#(4(5(0(0(x1:S))))) -> 0#(2(1(x1:S))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 4#(0(5(0(2(1(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 5#(0(2(1(x1:S)))) 1#(4(5(0(x1:S)))) -> 0#(2(1(2(5(x1:S))))) 1#(4(5(0(x1:S)))) -> 1#(2(5(x1:S))) 1#(4(5(0(x1:S)))) -> 4#(0(2(1(2(5(x1:S)))))) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 0#(3(1(2(5(4(x1:S)))))) 1#(4(5(2(0(x1:S))))) -> 1#(2(5(4(x1:S)))) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 0#(1(3(1(3(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 0#(3(1(3(x1:S)))) 1#(5(0(1(x1:S)))) -> 1#(5(0(3(1(3(x1:S)))))) 1#(5(0(1(x1:S)))) -> 1#(3(1(3(5(x1:S))))) 1#(5(0(1(x1:S)))) -> 1#(3(5(x1:S))) 1#(5(0(1(x1:S)))) -> 1#(3(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(0(3(1(3(x1:S))))) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 0#(5(1(5(x1:S)))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 1#(2(0(4(x1:S)))) -> 0#(x1:S) 1#(2(0(4(x1:S)))) -> 1#(3(4(2(0(x1:S))))) 1#(2(0(4(x1:S)))) -> 4#(2(0(x1:S))) 1#(2(0(5(0(x1:S))))) -> 0#(2(0(x1:S))) 1#(2(0(5(0(x1:S))))) -> 1#(0(2(0(x1:S)))) 1#(2(0(5(0(x1:S))))) -> 5#(2(1(0(2(0(x1:S)))))) 1#(3(0(4(x1:S)))) -> 0#(x1:S) 1#(3(0(4(x1:S)))) -> 1#(3(4(2(0(x1:S))))) 1#(3(0(4(x1:S)))) -> 4#(2(0(x1:S))) 4#(1(0(0(1(x1:S))))) -> 0#(4(0(3(1(x1:S))))) 4#(1(0(0(1(x1:S))))) -> 0#(3(1(x1:S))) 4#(1(0(0(1(x1:S))))) -> 1#(0(4(0(3(1(x1:S)))))) 4#(1(0(0(1(x1:S))))) -> 4#(0(3(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 4#(1(0(0(x1:S)))) -> 0#(2(4(2(0(1(x1:S)))))) 4#(1(0(0(x1:S)))) -> 0#(3(4(3(0(1(x1:S)))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(0(x1:S)))) -> 4#(2(0(1(x1:S)))) 4#(1(0(0(x1:S)))) -> 4#(3(0(1(x1:S)))) 4#(1(0(5(x1:S)))) -> 0#(2(1(5(x1:S)))) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(0(5(x1:S)))) -> 4#(0(2(1(5(x1:S))))) 4#(1(5(3(1(x1:S))))) -> 1#(5(4(3(1(x1:S))))) 4#(1(5(3(1(x1:S))))) -> 4#(3(1(x1:S))) 4#(1(5(3(1(x1:S))))) -> 5#(4(3(1(x1:S)))) 4#(1(2(0(0(x1:S))))) -> 0#(0(2(1(x1:S)))) 4#(1(2(0(0(x1:S))))) -> 0#(2(1(x1:S))) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(0(x1:S))))) -> 4#(2(0(0(2(1(x1:S)))))) 4#(1(2(0(x1:S)))) -> 0#(2(1(3(4(x1:S))))) 4#(1(2(0(x1:S)))) -> 1#(3(4(x1:S))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 0#(1(5(x1:S))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 1#(3(4(4(0(1(x1:S)))))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(4(0(5(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(5(x1:S)) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 0#(0(5(3(1(x1:S))))) 4#(5(0(1(0(x1:S))))) -> 0#(5(3(1(x1:S)))) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(0(1(0(x1:S))))) -> 4#(0(0(5(3(1(x1:S)))))) 4#(5(0(1(0(x1:S))))) -> 5#(3(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 0#(5(x1:S)) 4#(5(1(1(0(x1:S))))) -> 0#(2(1(5(1(x1:S))))) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 4#(0(2(1(5(1(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 4#(5(1(5(1(x1:S))))) -> 1#(5(5(3(1(x1:S))))) 4#(5(1(5(1(x1:S))))) -> 4#(1(5(5(3(1(x1:S)))))) 4#(5(1(5(1(x1:S))))) -> 5#(5(3(1(x1:S)))) 4#(5(1(5(1(x1:S))))) -> 5#(3(1(x1:S))) 5#(0(0(4(1(x1:S))))) -> 0#(1(x1:S)) 5#(0(0(4(1(x1:S))))) -> 0#(3(0(1(x1:S)))) 5#(0(0(4(1(x1:S))))) -> 4#(0(3(0(1(x1:S))))) 5#(0(0(4(1(x1:S))))) -> 5#(4(0(3(0(1(x1:S)))))) 5#(0(0(4(5(x1:S))))) -> 0#(0(3(5(5(x1:S))))) 5#(0(0(4(5(x1:S))))) -> 0#(5(5(x1:S))) 5#(0(0(4(5(x1:S))))) -> 0#(3(4(0(5(5(x1:S)))))) 5#(0(0(4(5(x1:S))))) -> 0#(3(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 4#(0(0(3(5(5(x1:S)))))) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(0(4(x1:S)))) -> 0#(2(0(x1:S))) 5#(0(0(4(x1:S)))) -> 0#(x1:S) 5#(0(0(4(x1:S)))) -> 4#(0(2(0(x1:S)))) 5#(0(0(4(x1:S)))) -> 5#(4(0(2(0(x1:S))))) 5#(0(1(4(x1:S)))) -> 0#(x1:S) 5#(0(1(4(x1:S)))) -> 1#(5(4(3(0(x1:S))))) 5#(0(1(4(x1:S)))) -> 4#(3(0(x1:S))) 5#(0(1(4(x1:S)))) -> 5#(4(3(0(x1:S)))) 5#(0(4(1(0(x1:S))))) -> 0#(4(0(3(1(x1:S))))) 5#(0(4(1(0(x1:S))))) -> 0#(3(1(x1:S))) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(0(4(1(0(x1:S))))) -> 4#(0(3(1(x1:S)))) 5#(0(4(1(0(x1:S))))) -> 5#(0(4(0(3(1(x1:S)))))) 5#(1(5(0(1(x1:S))))) -> 0#(3(5(1(1(x1:S))))) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(0(3(5(1(1(x1:S)))))) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 0#(1(5(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 1#(2(4(0(1(5(x1:S)))))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 4#(3(1(5(5(x1:S))))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 1#(3(4(0(x1:S)))) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) 5#(5(2(0(4(x1:S))))) -> 0#(x1:S) 5#(5(2(0(4(x1:S))))) -> 4#(2(0(x1:S))) 5#(5(2(0(4(x1:S))))) -> 5#(5(3(4(2(0(x1:S)))))) 5#(5(2(0(4(x1:S))))) -> 5#(3(4(2(0(x1:S))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(1(0(0(x1:S)))) -> 0#(0(1(x1:S))) 0#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 0#(1(0(0(x1:S)))) -> 1#(x1:S) 1#(0(0(4(x1:S)))) -> 0#(x1:S) 1#(0(1(0(x1:S)))) -> 1#(x1:S) 1#(0(1(4(5(x1:S))))) -> 0#(x1:S) 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 1#(3(1(5(4(0(x1:S)))))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 1#(2(4(x1:S))) 1#(0(4(0(4(x1:S))))) -> 0#(x1:S) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(0(5(0(x1:S)))) -> 0#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 0#(5(x1:S)) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 1#(2(5(x1:S))) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 1#(2(5(4(x1:S)))) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 0#(1(3(1(3(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 1#(3(1(3(5(x1:S))))) 1#(5(0(1(x1:S)))) -> 1#(3(5(x1:S))) 1#(5(0(1(x1:S)))) -> 1#(3(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 0#(5(1(5(x1:S)))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 1#(2(0(4(x1:S)))) -> 0#(x1:S) 1#(3(0(4(x1:S)))) -> 0#(x1:S) 4#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 1#(3(4(x1:S))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 0#(1(5(x1:S))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 1#(3(4(4(0(1(x1:S)))))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(4(0(5(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(5(x1:S)) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 0#(5(x1:S)) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(1(x1:S))))) -> 0#(1(x1:S)) 5#(0(0(4(5(x1:S))))) -> 0#(5(5(x1:S))) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(0(4(x1:S)))) -> 0#(x1:S) 5#(0(1(4(x1:S)))) -> 0#(x1:S) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 0#(1(5(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 1#(2(4(0(1(5(x1:S)))))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 1#(3(4(0(x1:S)))) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) 5#(5(2(0(4(x1:S))))) -> 0#(x1:S) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 0#(1(0(0(x1:S)))) -> 0#(0(1(x1:S))) 0#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 0#(1(0(0(x1:S)))) -> 1#(x1:S) 1#(0(0(4(x1:S)))) -> 0#(x1:S) 1#(0(1(0(x1:S)))) -> 1#(x1:S) 1#(0(1(4(5(x1:S))))) -> 0#(x1:S) 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 1#(3(1(5(4(0(x1:S)))))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 1#(2(4(x1:S))) 1#(0(4(0(4(x1:S))))) -> 0#(x1:S) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(0(5(0(x1:S)))) -> 0#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 0#(5(x1:S)) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 1#(2(5(x1:S))) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 1#(2(5(4(x1:S)))) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 0#(1(3(1(3(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 1#(3(1(3(5(x1:S))))) 1#(5(0(1(x1:S)))) -> 1#(3(5(x1:S))) 1#(5(0(1(x1:S)))) -> 1#(3(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 0#(5(1(5(x1:S)))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 1#(2(0(4(x1:S)))) -> 0#(x1:S) 1#(3(0(4(x1:S)))) -> 0#(x1:S) 4#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 1#(3(4(x1:S))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 0#(1(5(x1:S))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 1#(3(4(4(0(1(x1:S)))))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(4(0(5(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(5(x1:S)) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 0#(5(x1:S)) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(1(x1:S))))) -> 0#(1(x1:S)) 5#(0(0(4(5(x1:S))))) -> 0#(5(5(x1:S))) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(0(4(x1:S)))) -> 0#(x1:S) 5#(0(1(4(x1:S)))) -> 0#(x1:S) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 0#(1(5(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 1#(2(4(0(1(5(x1:S)))))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 1#(3(4(0(x1:S)))) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) 5#(5(2(0(4(x1:S))))) -> 0#(x1:S) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [0#](X) = 2.X + 1 [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 0#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 0#(1(0(0(x1:S)))) -> 1#(x1:S) 1#(0(0(4(x1:S)))) -> 0#(x1:S) 1#(0(1(0(x1:S)))) -> 1#(x1:S) 1#(0(1(4(5(x1:S))))) -> 0#(x1:S) 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 1#(3(1(5(4(0(x1:S)))))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 1#(2(4(x1:S))) 1#(0(4(0(4(x1:S))))) -> 0#(x1:S) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(0(5(0(x1:S)))) -> 0#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 0#(5(x1:S)) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 1#(2(5(x1:S))) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 1#(2(5(4(x1:S)))) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 0#(1(3(1(3(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 1#(3(1(3(5(x1:S))))) 1#(5(0(1(x1:S)))) -> 1#(3(5(x1:S))) 1#(5(0(1(x1:S)))) -> 1#(3(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 0#(5(1(5(x1:S)))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 1#(2(0(4(x1:S)))) -> 0#(x1:S) 1#(3(0(4(x1:S)))) -> 0#(x1:S) 4#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 1#(3(4(x1:S))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 0#(1(5(x1:S))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 1#(3(4(4(0(1(x1:S)))))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(4(0(5(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(5(x1:S)) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 0#(5(x1:S)) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(1(x1:S))))) -> 0#(1(x1:S)) 5#(0(0(4(5(x1:S))))) -> 0#(5(5(x1:S))) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(0(4(x1:S)))) -> 0#(x1:S) 5#(0(1(4(x1:S)))) -> 0#(x1:S) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 0#(1(5(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 1#(2(4(0(1(5(x1:S)))))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 1#(3(4(0(x1:S)))) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) 5#(5(2(0(4(x1:S))))) -> 0#(x1:S) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 0#(1(0(0(x1:S)))) -> 1#(x1:S) 1#(0(0(4(x1:S)))) -> 0#(x1:S) 1#(0(1(0(x1:S)))) -> 1#(x1:S) 1#(0(1(4(5(x1:S))))) -> 0#(x1:S) 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 1#(3(1(5(4(0(x1:S)))))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 1#(2(4(x1:S))) 1#(0(4(0(4(x1:S))))) -> 0#(x1:S) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(0(5(0(x1:S)))) -> 0#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 0#(5(x1:S)) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 1#(2(5(x1:S))) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 1#(2(5(4(x1:S)))) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 0#(1(3(1(3(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 1#(3(1(3(5(x1:S))))) 1#(5(0(1(x1:S)))) -> 1#(3(5(x1:S))) 1#(5(0(1(x1:S)))) -> 1#(3(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 0#(5(1(5(x1:S)))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 1#(2(0(4(x1:S)))) -> 0#(x1:S) 1#(3(0(4(x1:S)))) -> 0#(x1:S) 4#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 1#(3(4(x1:S))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 0#(1(5(x1:S))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 1#(3(4(4(0(1(x1:S)))))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(4(0(5(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(5(x1:S)) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 0#(5(x1:S)) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(1(x1:S))))) -> 0#(1(x1:S)) 5#(0(0(4(5(x1:S))))) -> 0#(5(5(x1:S))) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(0(4(x1:S)))) -> 0#(x1:S) 5#(0(1(4(x1:S)))) -> 0#(x1:S) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 0#(1(5(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 1#(2(4(0(1(5(x1:S)))))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 1#(3(4(0(x1:S)))) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) 5#(5(2(0(4(x1:S))))) -> 0#(x1:S) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 0#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 0#(1(0(0(x1:S)))) -> 1#(x1:S) 1#(0(0(4(x1:S)))) -> 0#(x1:S) 1#(0(1(0(x1:S)))) -> 1#(x1:S) 1#(0(1(4(5(x1:S))))) -> 0#(x1:S) 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 1#(3(1(5(4(0(x1:S)))))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 1#(2(4(x1:S))) 1#(0(4(0(4(x1:S))))) -> 0#(x1:S) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(0(5(0(x1:S)))) -> 0#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 0#(5(x1:S)) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 1#(2(5(x1:S))) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 1#(2(5(4(x1:S)))) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 0#(1(3(1(3(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 1#(3(1(3(5(x1:S))))) 1#(5(0(1(x1:S)))) -> 1#(3(5(x1:S))) 1#(5(0(1(x1:S)))) -> 1#(3(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 0#(5(1(5(x1:S)))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 1#(2(0(4(x1:S)))) -> 0#(x1:S) 1#(3(0(4(x1:S)))) -> 0#(x1:S) 4#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 1#(3(4(x1:S))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 0#(1(5(x1:S))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 1#(3(4(4(0(1(x1:S)))))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(4(0(5(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(5(x1:S)) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 0#(5(x1:S)) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(1(x1:S))))) -> 0#(1(x1:S)) 5#(0(0(4(5(x1:S))))) -> 0#(5(5(x1:S))) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(0(4(x1:S)))) -> 0#(x1:S) 5#(0(1(4(x1:S)))) -> 0#(x1:S) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 0#(1(5(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 1#(2(4(0(1(5(x1:S)))))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 1#(3(4(0(x1:S)))) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) 5#(5(2(0(4(x1:S))))) -> 0#(x1:S) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 0#(1(0(0(x1:S)))) -> 1#(x1:S) 1#(0(0(4(x1:S)))) -> 0#(x1:S) 1#(0(1(0(x1:S)))) -> 1#(x1:S) 1#(0(1(4(5(x1:S))))) -> 0#(x1:S) 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 1#(3(1(5(4(0(x1:S)))))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 1#(2(4(x1:S))) 1#(0(4(0(4(x1:S))))) -> 0#(x1:S) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(0(5(0(x1:S)))) -> 0#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 0#(5(x1:S)) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 1#(2(5(x1:S))) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 1#(2(5(4(x1:S)))) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 0#(1(3(1(3(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 1#(3(1(3(5(x1:S))))) 1#(5(0(1(x1:S)))) -> 1#(3(5(x1:S))) 1#(5(0(1(x1:S)))) -> 1#(3(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 0#(5(1(5(x1:S)))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 1#(2(0(4(x1:S)))) -> 0#(x1:S) 1#(3(0(4(x1:S)))) -> 0#(x1:S) 4#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 1#(3(4(x1:S))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 0#(1(5(x1:S))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 1#(3(4(4(0(1(x1:S)))))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(4(0(5(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(5(x1:S)) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 0#(5(x1:S)) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(1(x1:S))))) -> 0#(1(x1:S)) 5#(0(0(4(5(x1:S))))) -> 0#(5(5(x1:S))) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(0(4(x1:S)))) -> 0#(x1:S) 5#(0(1(4(x1:S)))) -> 0#(x1:S) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 0#(1(5(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 1#(2(4(0(1(5(x1:S)))))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 1#(3(4(0(x1:S)))) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) 5#(5(2(0(4(x1:S))))) -> 0#(x1:S) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(1(0(0(x1:S)))) -> 1#(x1:S) 1#(0(0(4(x1:S)))) -> 0#(x1:S) 1#(0(1(0(x1:S)))) -> 1#(x1:S) 1#(0(1(4(5(x1:S))))) -> 0#(x1:S) 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 1#(3(1(5(4(0(x1:S)))))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 1#(2(4(x1:S))) 1#(0(4(0(4(x1:S))))) -> 0#(x1:S) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(0(5(0(x1:S)))) -> 0#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 0#(5(x1:S)) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 1#(2(5(x1:S))) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 1#(2(5(4(x1:S)))) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 0#(1(3(1(3(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 1#(3(1(3(5(x1:S))))) 1#(5(0(1(x1:S)))) -> 1#(3(5(x1:S))) 1#(5(0(1(x1:S)))) -> 1#(3(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 0#(5(1(5(x1:S)))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 1#(2(0(4(x1:S)))) -> 0#(x1:S) 1#(3(0(4(x1:S)))) -> 0#(x1:S) 4#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 1#(3(4(x1:S))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 0#(1(5(x1:S))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 1#(3(4(4(0(1(x1:S)))))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(4(0(5(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(5(x1:S)) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 0#(5(x1:S)) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(1(x1:S))))) -> 0#(1(x1:S)) 5#(0(0(4(5(x1:S))))) -> 0#(5(5(x1:S))) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(0(4(x1:S)))) -> 0#(x1:S) 5#(0(1(4(x1:S)))) -> 0#(x1:S) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 0#(1(5(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 1#(2(4(0(1(5(x1:S)))))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 1#(3(4(0(x1:S)))) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) 5#(5(2(0(4(x1:S))))) -> 0#(x1:S) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 0#(1(0(0(x1:S)))) -> 1#(x1:S) 1#(0(0(4(x1:S)))) -> 0#(x1:S) 1#(0(1(0(x1:S)))) -> 1#(x1:S) 1#(0(1(4(5(x1:S))))) -> 0#(x1:S) 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 1#(3(1(5(4(0(x1:S)))))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 1#(2(4(x1:S))) 1#(0(4(0(4(x1:S))))) -> 0#(x1:S) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(0(5(0(x1:S)))) -> 0#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 0#(5(x1:S)) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 1#(2(5(x1:S))) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 1#(2(5(4(x1:S)))) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 0#(1(3(1(3(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 1#(3(1(3(5(x1:S))))) 1#(5(0(1(x1:S)))) -> 1#(3(5(x1:S))) 1#(5(0(1(x1:S)))) -> 1#(3(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 0#(5(1(5(x1:S)))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 1#(2(0(4(x1:S)))) -> 0#(x1:S) 1#(3(0(4(x1:S)))) -> 0#(x1:S) 4#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 1#(3(4(x1:S))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 0#(1(5(x1:S))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 1#(3(4(4(0(1(x1:S)))))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(4(0(5(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(5(x1:S)) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 0#(5(x1:S)) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(1(x1:S))))) -> 0#(1(x1:S)) 5#(0(0(4(5(x1:S))))) -> 0#(5(5(x1:S))) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(0(4(x1:S)))) -> 0#(x1:S) 5#(0(1(4(x1:S)))) -> 0#(x1:S) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 0#(1(5(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 1#(2(4(0(1(5(x1:S)))))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 1#(3(4(0(x1:S)))) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) 5#(5(2(0(4(x1:S))))) -> 0#(x1:S) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [0#](X) = 2.X + 2 [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 1#(0(0(4(x1:S)))) -> 0#(x1:S) 1#(0(1(0(x1:S)))) -> 1#(x1:S) 1#(0(1(4(5(x1:S))))) -> 0#(x1:S) 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 1#(3(1(5(4(0(x1:S)))))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(1(4(x1:S)))) -> 1#(2(4(x1:S))) 1#(0(4(0(4(x1:S))))) -> 0#(x1:S) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(0(5(0(x1:S)))) -> 0#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 0#(5(x1:S)) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 1#(2(5(x1:S))) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 1#(2(5(4(x1:S)))) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 0#(1(3(1(3(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 1#(3(1(3(5(x1:S))))) 1#(5(0(1(x1:S)))) -> 1#(3(5(x1:S))) 1#(5(0(1(x1:S)))) -> 1#(3(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 0#(5(1(5(x1:S)))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 1#(2(0(4(x1:S)))) -> 0#(x1:S) 1#(3(0(4(x1:S)))) -> 0#(x1:S) 4#(1(0(0(x1:S)))) -> 0#(1(x1:S)) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 1#(3(4(x1:S))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 0#(1(5(x1:S))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 1#(3(4(4(0(1(x1:S)))))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(4(0(5(x1:S)))) 4#(5(0(0(x1:S)))) -> 0#(5(x1:S)) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 0#(5(x1:S)) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(3(4(0(5(x1:S))))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(1(x1:S))))) -> 0#(1(x1:S)) 5#(0(0(4(5(x1:S))))) -> 0#(5(5(x1:S))) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(0(4(x1:S)))) -> 0#(x1:S) 5#(0(1(4(x1:S)))) -> 0#(x1:S) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 0#(1(5(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 1#(2(4(0(1(5(x1:S)))))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 1#(3(4(0(x1:S)))) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) 5#(5(2(0(4(x1:S))))) -> 0#(x1:S) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(0(1(0(x1:S)))) -> 1#(x1:S) 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) ->rm: cannot remove ‘/tmp/muTerm14115496761843993368.in.dec’: No such file or directory 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(0(1(0(x1:S)))) -> 1#(x1:S) 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = X [2](X) = X [3](X) = X [1#](X) = X + 2 [4#](X) = X + 2 [5#](X) = X + 1 Problem 1: SCC Processor: -> Pairs: 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reductrm: cannot remove ‘/tmp/muTerm14115496761843993368.in.form’: No such file or directory ion Pair Processor: -> Pairs: 1#(0(1(4(5(x1:S))))) -> 1#(5(4(0(x1:S)))) 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(0(1(4(5(x1:S))))) -> 4#(0(x1:S)) 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = X [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X Problem 1: SCC Processor: -> Pairs: 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(0(1(4(5(x1:S))))) -> 5#(4(0(x1:S))) 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = X [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(0(4(0(4(x1:S))))) -> 4#(0(x1:S)) 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = 2.X [4#](X) = 2.X [5#](X) = 2.X Problem 1: SCC Processor: -> Pairs: 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(1(0(0(x1:S))))) -> 1#(x1:S) 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 4#(0(5(x1:S))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = X [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(1(4(5(0(x1:S))))) -> 5#(x1:S) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(1(0(x1:S)))) -> 1#(x1:S) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = X [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = X [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(0(x1:S)))) -> 5#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = X + 1 [4#](X) = X + 1 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(2(0(x1:S))))) -> 4#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 1#(5(0(4(5(x1:S))))) -> 5#(1(5(x1:S))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = X + 1 Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(0(0(x1:S))))) -> 1#(x1:S) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(4(5(2(0(x1:S))))) -> 5#(4(x1:S)) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = X [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(1(x1:S)))) -> 5#(x1:S) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 1#(5(x1:S)) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = X + 1 [4#](X) = X + 1 [5#](X) = 2.X Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(0(x1:S)))) -> 1#(x1:S) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = X [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = X [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(0(5(x1:S)))) -> 1#(5(x1:S)) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = X + 2 [4#](X) = X + 2 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(2(0(0(x1:S))))) -> 1#(x1:S) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = 2.X + 1 [4#](X) = 2.X + 1 [5#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(2(0(x1:S)))) -> 4#(x1:S) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [1#](X) = 2.X + 2 [4#](X) = 2.X + 2 [5#](X) = 2.X Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 1#(5(0(4(5(x1:S))))) -> 4#(0(5(1(5(x1:S))))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 0 [1](X) = 2 [4](X) = 2 [5](X) = 2 [2](X) = 0 [3](X) = 1 [1#](X) = 2 [4#](X) = 2.X + 1 [5#](X) = 2 Problem 1: SCC Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(1(3(0(5(x1:S))))) -> 1#(5(x1:S)) 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(0(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) 4#(5(1(1(0(x1:S))))) -> 1#(5(1(x1:S))) 4#(5(1(1(0(x1:S))))) -> 1#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(0(4(1(0(x1:S))))) -> 1#(x1:S) 5#(1(5(0(1(x1:S))))) -> 1#(1(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 1#(5(x1:S)) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 1#(5(5(x1:S))) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->->Cycle: ->->-> Pairs: 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) The problem is decomposed in 2 subproblems. Problem 1.1: Reduction Pair Processor: -> Pairs: 1#(1(4(5(0(x1:S))))) -> 1#(1(3(4(0(5(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(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) = 2.X + 2 [2](X) = 0 [3](X) = 0 [1#](X) = X Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pair Processor: -> Pairs: 4#(1(3(0(5(x1:S))))) -> 4#(0(1(5(x1:S)))) 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 0 [1](X) = 2 [4](X) = 2 [5](X) = 2 [2](X) = 0 [3](X) = 0 [4#](X) = 2.X + 2 [5#](X) = 2 Problem 1.2: SCC Processor: -> Pairs: 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 4#(4(1(0(1(x1:S))))) -> 4#(0(1(x1:S))) 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 0 [1](X) = 2 [4](X) = 2 [5](X) = 2 [2](X) = X [3](X) = 2 [4#](X) = X + 1 [5#](X) = 1 Problem 1.2: SCC Processor: -> Pairs: 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 4#(4(1(0(1(x1:S))))) -> 4#(4(0(1(x1:S)))) 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X + 2 [1](X) = 2.X [4](X) = 2.X + 2 [5](X) = 2.X [2](X) = 0 [3](X) = 0 [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1.2: SCC Processor: -> Pairs: 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 4#(5(0(0(x1:S)))) -> 4#(0(5(x1:S))) 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 1 [1](X) = 2.X + 1 [4](X) = 2.X + 1 [5](X) = X [2](X) = X [3](X) = 0 [4#](X) = 2.X [5#](X) = 2.X + 1 Problem 1.2: SCC Processor: -> Pairs: 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 4#(5(0(0(x1:S)))) -> 5#(x1:S) 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 0 [1](X) = 2 [4](X) = 2.X [5](X) = 2.X + 2 [2](X) = 0 [3](X) = 2.X [4#](X) = X [5#](X) = 0 Problem 1.2: SCC Processor: -> Pairs: 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 4#(5(1(1(0(x1:S))))) -> 4#(0(5(x1:S))) 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2 [1](X) = 2.X + 2 [4](X) = X [5](X) = 2.X + 2 [2](X) = 2 [3](X) = 0 [4#](X) = X [5#](X) = 2 Problem 1.2: SCC Processor: -> Pairs: 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 4#(5(1(1(0(x1:S))))) -> 5#(1(x1:S)) 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 1 [5](X) = 2.X + 2 [2](X) = 0 [3](X) = 0 [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1.2: SCC Processor: -> Pairs: 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 4#(5(1(1(0(x1:S))))) -> 5#(x1:S) 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(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 + 2 [5](X) = 2.X + 2 [2](X) = X [3](X) = X [4#](X) = 2.X + 2 [5#](X) = 2.X + 2 Problem 1.2: SCC Processor: -> Pairs: 5#(0(0(4(5(x1:S))))) -> 4#(0(5(5(x1:S)))) 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 4#(0(1(5(x1:S)))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 4#(0(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 5#(0(0(4(5(x1:S))))) -> 5#(5(x1:S)) 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = X [2](X) = 0 [3](X) = 0 [5#](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 5#(1(5(0(1(x1:S))))) -> 5#(1(1(x1:S))) 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 1 [1](X) = X [4](X) = 2.X + 1 [5](X) = 2.X + 1 [2](X) = X [3](X) = X [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 5#(4(1(1(0(x1:S))))) -> 5#(x1:S) 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [4](X) = 2.X + 2 [5](X) = 2.X [2](X) = 0 [3](X) = 0 [5#](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 5#(4(1(5(x1:S)))) -> 5#(5(x1:S)) 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2 [1](X) = 2.X + 2 [4](X) = X [5](X) = 2.X + 2 [2](X) = 0 [3](X) = 0 [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 5#(4(5(1(0(x1:S))))) -> 5#(1(3(4(0(x1:S))))) 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = X + 2 [4](X) = 2.X + 1 [5](X) = 2.X + 2 [2](X) = 0 [3](X) = 0 [5#](X) = X Problem 1.2: SCC Processor: -> Pairs: 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) ->->-> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) Problem 1.2: Reduction Pair Processor: -> Pairs: 5#(4(5(1(0(x1:S))))) -> 5#(5(1(3(4(0(x1:S)))))) -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) -> Usable rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2 [1](X) = 2.X + 2 [4](X) = X [5](X) = 2.X + 2 [2](X) = 1 [3](X) = 0 [5#](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: 0(1(0(0(x1:S)))) -> 2(0(2(0(0(1(x1:S)))))) 1(0(0(1(x1:S)))) -> 1(0(3(0(1(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(2(0(x1:S))))) 1(0(0(4(x1:S)))) -> 1(4(0(3(2(0(x1:S)))))) 1(0(1(0(x1:S)))) -> 1(3(0(0(3(1(x1:S)))))) 1(0(1(4(5(x1:S))))) -> 1(3(1(5(4(0(x1:S)))))) 1(0(1(4(x1:S)))) -> 1(0(3(1(2(4(x1:S)))))) 1(0(1(5(x1:S)))) -> 1(0(2(1(5(x1:S))))) 1(0(4(0(4(x1:S))))) -> 1(0(4(3(4(0(x1:S)))))) 1(0(4(1(x1:S)))) -> 1(0(3(4(1(x1:S))))) 1(0(5(0(x1:S)))) -> 1(2(5(2(0(0(x1:S)))))) 1(1(1(0(0(x1:S))))) -> 1(1(0(0(3(1(x1:S)))))) 1(1(4(5(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 1(4(1(0(x1:S)))) -> 1(4(0(2(1(x1:S))))) 1(4(1(0(x1:S)))) -> 1(4(0(3(1(x1:S))))) 1(4(5(0(0(x1:S))))) -> 4(0(5(0(2(1(x1:S)))))) 1(4(5(0(x1:S)))) -> 4(0(2(1(2(5(x1:S)))))) 1(4(5(2(0(x1:S))))) -> 0(3(1(2(5(4(x1:S)))))) 1(5(0(1(x1:S)))) -> 0(1(3(1(3(5(x1:S)))))) 1(5(0(1(x1:S)))) -> 1(5(0(3(1(3(x1:S)))))) 1(5(0(4(5(x1:S))))) -> 3(4(0(5(1(5(x1:S)))))) 1(2(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 1(2(0(5(0(x1:S))))) -> 5(2(1(0(2(0(x1:S)))))) 1(3(0(4(x1:S)))) -> 1(3(4(2(0(x1:S))))) 4(1(0(0(1(x1:S))))) -> 1(0(4(0(3(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(2(4(2(0(1(x1:S)))))) 4(1(0(0(x1:S)))) -> 0(3(4(3(0(1(x1:S)))))) 4(1(0(5(x1:S)))) -> 4(0(2(1(5(x1:S))))) 4(1(5(3(1(x1:S))))) -> 2(1(5(4(3(1(x1:S)))))) 4(1(2(0(0(x1:S))))) -> 4(2(0(0(2(1(x1:S)))))) 4(1(2(0(x1:S)))) -> 3(0(2(1(3(4(x1:S)))))) 4(1(3(0(5(x1:S))))) -> 3(2(4(0(1(5(x1:S)))))) 4(4(1(0(1(x1:S))))) -> 1(3(4(4(0(1(x1:S)))))) 4(5(0(0(x1:S)))) -> 3(0(4(0(5(x1:S))))) 4(5(0(1(0(x1:S))))) -> 4(0(0(5(3(1(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 1(1(3(4(0(5(x1:S)))))) 4(5(1(1(0(x1:S))))) -> 4(0(2(1(5(1(x1:S)))))) 4(5(1(5(1(x1:S))))) -> 4(1(5(5(3(1(x1:S)))))) 4(5(5(0(4(x1:S))))) -> 5(4(0(5(4(2(x1:S)))))) 5(0(0(4(1(x1:S))))) -> 5(4(0(3(0(1(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 0(3(4(0(5(5(x1:S)))))) 5(0(0(4(5(x1:S))))) -> 4(0(0(3(5(5(x1:S)))))) 5(0(0(4(x1:S)))) -> 5(4(0(2(0(x1:S))))) 5(0(1(4(x1:S)))) -> 1(5(4(3(0(x1:S))))) 5(0(4(1(0(x1:S))))) -> 5(0(4(0(3(1(x1:S)))))) 5(1(5(0(1(x1:S))))) -> 5(0(3(5(1(1(x1:S)))))) 5(4(1(1(0(x1:S))))) -> 1(2(4(0(1(5(x1:S)))))) 5(4(1(5(x1:S)))) -> 4(3(1(5(5(x1:S))))) 5(4(5(1(0(x1:S))))) -> 5(5(1(3(4(0(x1:S)))))) 5(5(2(0(4(x1:S))))) -> 5(5(3(4(2(0(x1:S)))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite.