YES Problem 1: (VAR v_NonEmpty:S x1:S) (RULES 0(2(1(2(x1:S)))) -> 0(2(2(1(5(x1:S))))) 0(2(4(3(1(x1:S))))) -> 1(3(4(2(3(0(x1:S)))))) 0(2(4(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 0(2(5(1(x1:S)))) -> 1(1(5(0(2(1(x1:S)))))) 0(2(5(3(1(x1:S))))) -> 5(0(2(3(5(1(x1:S)))))) 0(1(2(1(x1:S)))) -> 3(1(4(0(2(1(x1:S)))))) 0(1(2(x1:S))) -> 0(2(1(0(x1:S)))) 0(1(2(x1:S))) -> 0(2(4(1(5(x1:S))))) 0(1(2(x1:S))) -> 0(1(3(4(2(3(x1:S)))))) 0(1(2(x1:S))) -> 0(3(2(1(0(x1:S))))) 0(1(2(x1:S))) -> 1(0(2(3(x1:S)))) 0(1(2(x1:S))) -> 1(0(3(2(3(x1:S))))) 0(1(3(0(1(x1:S))))) -> 0(3(1(0(1(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 1(0(3(4(2(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(1(5(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(5(2(x1:S))) -> 0(2(4(5(3(x1:S))))) 0(5(2(x1:S))) -> 5(4(2(3(0(4(x1:S)))))) 0(5(3(1(x1:S)))) -> 0(1(4(4(3(5(x1:S)))))) 0(5(5(2(x1:S)))) -> 5(4(2(3(5(0(x1:S)))))) 2(0(1(2(x1:S)))) -> 0(2(3(2(1(1(x1:S)))))) 2(0(1(2(x1:S)))) -> 4(0(2(1(1(2(x1:S)))))) 2(0(1(x1:S))) -> 0(2(2(3(4(1(x1:S)))))) 2(0(1(x1:S))) -> 0(2(1(1(4(x1:S))))) 2(0(1(x1:S))) -> 0(3(2(3(1(1(x1:S)))))) 2(0(1(x1:S))) -> 0(3(2(4(1(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(4(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(x1:S)))) 2(0(1(x1:S))) -> 4(0(4(2(1(4(x1:S)))))) 2(0(4(1(x1:S)))) -> 3(0(2(4(1(x1:S))))) 2(0(5(1(x1:S)))) -> 5(4(2(1(0(x1:S))))) 2(0(5(4(1(x1:S))))) -> 0(4(5(3(2(1(x1:S)))))) 2(2(0(1(2(x1:S))))) -> 2(4(0(2(2(1(x1:S)))))) 2(2(5(1(x1:S)))) -> 3(2(2(4(5(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 1(0(2(4(4(x1:S))))) 2(4(0(1(x1:S)))) -> 3(0(0(2(4(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 5(4(0(2(1(1(x1:S)))))) 2(4(0(5(1(x1:S))))) -> 1(4(5(0(4(2(x1:S)))))) 2(4(2(3(1(x1:S))))) -> 4(2(2(3(3(1(x1:S)))))) 2(5(2(0(1(x1:S))))) -> 0(2(4(1(5(2(x1:S)))))) 2(5(2(1(x1:S)))) -> 1(5(2(2(3(1(x1:S)))))) 2(5(1(x1:S))) -> 0(2(1(5(1(x1:S))))) 2(5(1(x1:S))) -> 1(4(5(4(2(x1:S))))) 2(5(1(x1:S))) -> 1(5(0(2(4(1(x1:S)))))) 2(5(1(x1:S))) -> 5(0(2(1(4(x1:S))))) 2(5(1(x1:S))) -> 5(2(1(1(1(1(x1:S)))))) 2(5(1(x1:S))) -> 5(2(1(4(1(x1:S))))) 2(5(4(1(x1:S)))) -> 4(5(2(1(4(4(x1:S)))))) 2(5(5(2(x1:S)))) -> 5(5(2(3(2(x1:S))))) 2(5(5(1(x1:S)))) -> 1(5(4(2(4(5(x1:S)))))) ) Problem 1: Dependency Pairs Processor: -> Pairs: 0#(2(4(3(1(x1:S))))) -> 0#(3(2(1(0(x1:S))))) 0#(2(4(3(1(x1:S))))) -> 0#(x1:S) 0#(2(4(3(1(x1:S))))) -> 2#(1(0(x1:S))) 0#(2(4(3(1(x1:S))))) -> 2#(3(0(x1:S))) 0#(2(5(1(x1:S)))) -> 0#(2(1(x1:S))) 0#(1(2(1(x1:S)))) -> 0#(2(1(x1:S))) 0#(1(2(x1:S))) -> 0#(2(1(0(x1:S)))) 0#(1(2(x1:S))) -> 0#(3(2(1(0(x1:S))))) 0#(1(2(x1:S))) -> 0#(x1:S) 0#(1(2(x1:S))) -> 2#(1(0(x1:S))) 0#(5(5(2(x1:S)))) -> 0#(x1:S) 0#(5(5(2(x1:S)))) -> 2#(3(5(0(x1:S)))) 2#(0(1(2(x1:S)))) -> 0#(2(1(1(2(x1:S))))) 2#(0(1(2(x1:S)))) -> 2#(1(1(2(x1:S)))) 2#(0(1(x1:S))) -> 0#(2(1(x1:S))) 2#(0(5(1(x1:S)))) -> 0#(x1:S) 2#(0(5(1(x1:S)))) -> 2#(1(0(x1:S))) 2#(4(0(5(1(x1:S))))) -> 0#(4(2(x1:S))) 2#(4(0(5(1(x1:S))))) -> 2#(x1:S) 2#(5(2(0(1(x1:S))))) -> 0#(2(4(1(5(2(x1:S)))))) 2#(5(2(0(1(x1:S))))) -> 2#(4(1(5(2(x1:S))))) 2#(5(2(0(1(x1:S))))) -> 2#(x1:S) 2#(5(1(x1:S))) -> 2#(x1:S) 2#(5(5(2(x1:S)))) -> 2#(3(2(x1:S))) -> Rules: 0(2(1(2(x1:S)))) -> 0(2(2(1(5(x1:S))))) 0(2(4(3(1(x1:S))))) -> 1(3(4(2(3(0(x1:S)))))) 0(2(4(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 0(2(5(1(x1:S)))) -> 1(1(5(0(2(1(x1:S)))))) 0(2(5(3(1(x1:S))))) -> 5(0(2(3(5(1(x1:S)))))) 0(1(2(1(x1:S)))) -> 3(1(4(0(2(1(x1:S)))))) 0(1(2(x1:S))) -> 0(2(1(0(x1:S)))) 0(1(2(x1:S))) -> 0(2(4(1(5(x1:S))))) 0(1(2(x1:S))) -> 0(1(3(4(2(3(x1:S)))))) 0(1(2(x1:S))) -> 0(3(2(1(0(x1:S))))) 0(1(2(x1:S))) -> 1(0(2(3(x1:S)))) 0(1(2(x1:S))) -> 1(0(3(2(3(x1:S))))) 0(1(3(0(1(x1:S))))) -> 0(3(1(0(1(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 1(0(3(4(2(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(1(5(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(5(2(x1:S))) -> 0(2(4(5(3(x1:S))))) 0(5(2(x1:S))) -> 5(4(2(3(0(4(x1:S)))))) 0(5(3(1(x1:S)))) -> 0(1(4(4(3(5(x1:S)))))) 0(5(5(2(x1:S)))) -> 5(4(2(3(5(0(x1:S)))))) 2(0(1(2(x1:S)))) -> 0(2(3(2(1(1(x1:S)))))) 2(0(1(2(x1:S)))) -> 4(0(2(1(1(2(x1:S)))))) 2(0(1(x1:S))) -> 0(2(2(3(4(1(x1:S)))))) 2(0(1(x1:S))) -> 0(2(1(1(4(x1:S))))) 2(0(1(x1:S))) -> 0(3(2(3(1(1(x1:S)))))) 2(0(1(x1:S))) -> 0(3(2(4(1(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(4(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(x1:S)))) 2(0(1(x1:S))) -> 4(0(4(2(1(4(x1:S)))))) 2(0(4(1(x1:S)))) -> 3(0(2(4(1(x1:S))))) 2(0(5(1(x1:S)))) -> 5(4(2(1(0(x1:S))))) 2(0(5(4(1(x1:S))))) -> 0(4(5(3(2(1(x1:S)))))) 2(2(0(1(2(x1:S))))) -> 2(4(0(2(2(1(x1:S)))))) 2(2(5(1(x1:S)))) -> 3(2(2(4(5(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 1(0(2(4(4(x1:S))))) 2(4(0(1(x1:S)))) -> 3(0(0(2(4(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 5(4(0(2(1(1(x1:S)))))) 2(4(0(5(1(x1:S))))) -> 1(4(5(0(4(2(x1:S)))))) 2(4(2(3(1(x1:S))))) -> 4(2(2(3(3(1(x1:S)))))) 2(5(2(0(1(x1:S))))) -> 0(2(4(1(5(2(x1:S)))))) 2(5(2(1(x1:S)))) -> 1(5(2(2(3(1(x1:S)))))) 2(5(1(x1:S))) -> 0(2(1(5(1(x1:S))))) 2(5(1(x1:S))) -> 1(4(5(4(2(x1:S))))) 2(5(1(x1:S))) -> 1(5(0(2(4(1(x1:S)))))) 2(5(1(x1:S))) -> 5(0(2(1(4(x1:S))))) 2(5(1(x1:S))) -> 5(2(1(1(1(1(x1:S)))))) 2(5(1(x1:S))) -> 5(2(1(4(1(x1:S))))) 2(5(4(1(x1:S)))) -> 4(5(2(1(4(4(x1:S)))))) 2(5(5(2(x1:S)))) -> 5(5(2(3(2(x1:S))))) 2(5(5(1(x1:S)))) -> 1(5(4(2(4(5(x1:S)))))) Problem 1: SCC Processor: -> Pairs: 0#(2(4(3(1(x1:S))))) -> 0#(3(2(1(0(x1:S))))) 0#(2(4(3(1(x1:S))))) -> 0#(x1:S) 0#(2(4(3(1(x1:S))))) -> 2#(1(0(x1:S))) 0#(2(4(3(1(x1:S))))) -> 2#(3(0(x1:S))) 0#(2(5(1(x1:S)))) -> 0#(2(1(x1:S))) 0#(1(2(1(x1:S)))) -> 0#(2(1(x1:S))) 0#(1(2(x1:S))) -> 0#(2(1(0(x1:S)))) 0#(1(2(x1:S))) -> 0#(3(2(1(0(x1:S))))) 0#(1(2(x1:S))) -> 0#(x1:S) 0#(1(2(x1:S))) -> 2#(1(0(x1:S))) 0#(5(5(2(x1:S)))) -> 0#(x1:S) 0#(5(5(2(x1:S)))) -> 2#(3(5(0(x1:S)))) 2#(0(1(2(x1:S)))) -> 0#(2(1(1(2(x1:S))))) 2#(0(1(2(x1:S)))) -> 2#(1(1(2(x1:S)))) 2#(0(1(x1:S))) -> 0#(2(1(x1:S))) 2#(0(5(1(x1:S)))) -> 0#(x1:S) 2#(0(5(1(x1:S)))) -> 2#(1(0(x1:S))) 2#(4(0(5(1(x1:S))))) -> 0#(4(2(x1:S))) 2#(4(0(5(1(x1:S))))) -> 2#(x1:S) 2#(5(2(0(1(x1:S))))) -> 0#(2(4(1(5(2(x1:S)))))) 2#(5(2(0(1(x1:S))))) -> 2#(4(1(5(2(x1:S))))) 2#(5(2(0(1(x1:S))))) -> 2#(x1:S) 2#(5(1(x1:S))) -> 2#(x1:S) 2#(5(5(2(x1:S)))) -> 2#(3(2(x1:S))) -> Rules: 0(2(1(2(x1:S)))) -> 0(2(2(1(5(x1:S))))) 0(2(4(3(1(x1:S))))) -> 1(3(4(2(3(0(x1:S)))))) 0(2(4(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 0(2(5(1(x1:S)))) -> 1(1(5(0(2(1(x1:S)))))) 0(2(5(3(1(x1:S))))) -> 5(0(2(3(5(1(x1:S)))))) 0(1(2(1(x1:S)))) -> 3(1(4(0(2(1(x1:S)))))) 0(1(2(x1:S))) -> 0(2(1(0(x1:S)))) 0(1(2(x1:S))) -> 0(2(4(1(5(x1:S))))) 0(1(2(x1:S))) -> 0(1(3(4(2(3(x1:S)))))) 0(1(2(x1:S))) -> 0(3(2(1(0(x1:S))))) 0(1(2(x1:S))) -> 1(0(2(3(x1:S)))) 0(1(2(x1:S))) -> 1(0(3(2(3(x1:S))))) 0(1(3(0(1(x1:S))))) -> 0(3(1(0(1(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 1(0(3(4(2(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(1(5(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(5(2(x1:S))) -> 0(2(4(5(3(x1:S))))) 0(5(2(x1:S))) -> 5(4(2(3(0(4(x1:S)))))) 0(5(3(1(x1:S)))) -> 0(1(4(4(3(5(x1:S)))))) 0(5(5(2(x1:S)))) -> 5(4(2(3(5(0(x1:S)))))) 2(0(1(2(x1:S)))) -> 0(2(3(2(1(1(x1:S)))))) 2(0(1(2(x1:S)))) -> 4(0(2(1(1(2(x1:S)))))) 2(0(1(x1:S))) -> 0(2(2(3(4(1(x1:S)))))) 2(0(1(x1:S))) -> 0(2(1(1(4(x1:S))))) 2(0(1(x1:S))) -> 0(3(2(3(1(1(x1:S)))))) 2(0(1(x1:S))) -> 0(3(2(4(1(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(4(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(x1:S)))) 2(0(1(x1:S))) -> 4(0(4(2(1(4(x1:S)))))) 2(0(4(1(x1:S)))) -> 3(0(2(4(1(x1:S))))) 2(0(5(1(x1:S)))) -> 5(4(2(1(0(x1:S))))) 2(0(5(4(1(x1:S))))) -> 0(4(5(3(2(1(x1:S)))))) 2(2(0(1(2(x1:S))))) -> 2(4(0(2(2(1(x1:S)))))) 2(2(5(1(x1:S)))) -> 3(2(2(4(5(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 1(0(2(4(4(x1:S))))) 2(4(0(1(x1:S)))) -> 3(0(0(2(4(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 5(4(0(2(1(1(x1:S)))))) 2(4(0(5(1(x1:S))))) -> 1(4(5(0(4(2(x1:S)))))) 2(4(2(3(1(x1:S))))) -> 4(2(2(3(3(1(x1:S)))))) 2(5(2(0(1(x1:S))))) -> 0(2(4(1(5(2(x1:S)))))) 2(5(2(1(x1:S)))) -> 1(5(2(2(3(1(x1:S)))))) 2(5(1(x1:S))) -> 0(2(1(5(1(x1:S))))) 2(5(1(x1:S))) -> 1(4(5(4(2(x1:S))))) 2(5(1(x1:S))) -> 1(5(0(2(4(1(x1:S)))))) 2(5(1(x1:S))) -> 5(0(2(1(4(x1:S))))) 2(5(1(x1:S))) -> 5(2(1(1(1(1(x1:S)))))) 2(5(1(x1:S))) -> 5(2(1(4(1(x1:S))))) 2(5(4(1(x1:S)))) -> 4(5(2(1(4(4(x1:S)))))) 2(5(5(2(x1:S)))) -> 5(5(2(3(2(x1:S))))) 2(5(5(1(x1:S)))) -> 1(5(4(2(4(5(x1:S)))))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 0#(2(4(3(1(x1:S))))) -> 0#(x1:S) 0#(1(2(x1:S))) -> 0#(x1:S) 0#(5(5(2(x1:S)))) -> 0#(x1:S) ->->-> Rules: 0(2(1(2(x1:S)))) -> 0(2(2(1(5(x1:S))))) 0(2(4(3(1(x1:S))))) -> 1(3(4(2(3(0(x1:S)))))) 0(2(4(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 0(2(5(1(x1:S)))) -> 1(1(5(0(2(1(x1:S)))))) 0(2(5(3(1(x1:S))))) -> 5(0(2(3(5(1(x1:S)))))) 0(1(2(1(x1:S)))) -> 3(1(4(0(2(1(x1:S)))))) 0(1(2(x1:S))) -> 0(2(1(0(x1:S)))) 0(1(2(x1:S))) -> 0(2(4(1(5(x1:S))))) 0(1(2(x1:S))) -> 0(1(3(4(2(3(x1:S)))))) 0(1(2(x1:S))) -> 0(3(2(1(0(x1:S))))) 0(1(2(x1:S))) -> 1(0(2(3(x1:S)))) 0(1(2(x1:S))) -> 1(0(3(2(3(x1:S))))) 0(1(3(0(1(x1:S))))) -> 0(3(1(0(1(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 1(0(3(4(2(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(1(5(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(5(2(x1:S))) -> 0(2(4(5(3(x1:S))))) 0(5(2(x1:S))) -> 5(4(2(3(0(4(x1:S)))))) 0(5(3(1(x1:S)))) -> 0(1(4(4(3(5(x1:S)))))) 0(5(5(2(x1:S)))) -> 5(4(2(3(5(0(x1:S)))))) 2(0(1(2(x1:S)))) -> 0(2(3(2(1(1(x1:S)))))) 2(0(1(2(x1:S)))) -> 4(0(2(1(1(2(x1:S)))))) 2(0(1(x1:S))) -> 0(2(2(3(4(1(x1:S)))))) 2(0(1(x1:S))) -> 0(2(1(1(4(x1:S))))) 2(0(1(x1:S))) -> 0(3(2(3(1(1(x1:S)))))) 2(0(1(x1:S))) -> 0(3(2(4(1(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(4(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(x1:S)))) 2(0(1(x1:S))) -> 4(0(4(2(1(4(x1:S)))))) 2(0(4(1(x1:S)))) -> 3(0(2(4(1(x1:S))))) 2(0(5(1(x1:S)))) -> 5(4(2(1(0(x1:S))))) 2(0(5(4(1(x1:S))))) -> 0(4(5(3(2(1(x1:S)))))) 2(2(0(1(2(x1:S))))) -> 2(4(0(2(2(1(x1:S)))))) 2(2(5(1(x1:S)))) -> 3(2(2(4(5(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 1(0(2(4(4(x1:S))))) 2(4(0(1(x1:S)))) -> 3(0(0(2(4(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 5(4(0(2(1(1(x1:S)))))) 2(4(0(5(1(x1:S))))) -> 1(4(5(0(4(2(x1:S)))))) 2(4(2(3(1(x1:S))))) -> 4(2(2(3(3(1(x1:S)))))) 2(5(2(0(1(x1:S))))) -> 0(2(4(1(5(2(x1:S)))))) 2(5(2(1(x1:S)))) -> 1(5(2(2(3(1(x1:S)))))) 2(5(1(x1:S))) -> 0(2(1(5(1(x1:S))))) 2(5(1(x1:S))) -> 1(4(5(4(2(x1:S))))) 2(5(1(x1:S))) -> 1(5(0(2(4(1(x1:S)))))) 2(5(1(x1:S))) -> 5(0(2(1(4(x1:S))))) 2(5(1(x1:S))) -> 5(2(1(1(1(1(x1:S)))))) 2(5(1(x1:S))) -> 5(2(1(4(1(x1:S))))) 2(5(4(1(x1:S)))) -> 4(5(2(1(4(4(x1:S)))))) 2(5(5(2(x1:S)))) -> 5(5(2(3(2(x1:S))))) 2(5(5(1(x1:S)))) -> 1(5(4(2(4(5(x1:S)))))) ->->Cycle: ->->-> Pairs: 2#(4(0(5(1(x1:S))))) -> 2#(x1:S) 2#(5(2(0(1(x1:S))))) -> 2#(x1:S) 2#(5(1(x1:S))) -> 2#(x1:S) ->->-> Rules: 0(2(1(2(x1:S)))) -> 0(2(2(1(5(x1:S))))) 0(2(4(3(1(x1:S))))) -> 1(3(4(2(3(0(x1:S)))))) 0(2(4(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 0(2(5(1(x1:S)))) -> 1(1(5(0(2(1(x1:S)))))) 0(2(5(3(1(x1:S))))) -> 5(0(2(3(5(1(x1:S)))))) 0(1(2(1(x1:S)))) -> 3(1(4(0(2(1(x1:S)))))) 0(1(2(x1:S))) -> 0(2(1(0(x1:S)))) 0(1(2(x1:S))) -> 0(2(4(1(5(x1:S))))) 0(1(2(x1:S))) -> 0(1(3(4(2(3(x1:S)))))) 0(1(2(x1:S))) -> 0(3(2(1(0(x1:S))))) 0(1(2(x1:S))) -> 1(0(2(3(x1:S)))) 0(1(2(x1:S))) -> 1(0(3(2(3(x1:S))))) 0(1(3(0(1(x1:S))))) -> 0(3(1(0(1(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 1(0(3(4(2(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(1(5(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(5(2(x1:S))) -> 0(2(4(5(3(x1:S))))) 0(5(2(x1:S))) -> 5(4(2(3(0(4(x1:S)))))) 0(5(3(1(x1:S)))) -> 0(1(4(4(3(5(x1:S)))))) 0(5(5(2(x1:S)))) -> 5(4(2(3(5(0(x1:S)))))) 2(0(1(2(x1:S)))) -> 0(2(3(2(1(1(x1:S)))))) 2(0(1(2(x1:S)))) -> 4(0(2(1(1(2(x1:S)))))) 2(0(1(x1:S))) -> 0(2(2(3(4(1(x1:S)))))) 2(0(1(x1:S))) -> 0(2(1(1(4(x1:S))))) 2(0(1(x1:S))) -> 0(3(2(3(1(1(x1:S)))))) 2(0(1(x1:S))) -> 0(3(2(4(1(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(4(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(x1:S)))) 2(0(1(x1:S))) -> 4(0(4(2(1(4(x1:S)))))) 2(0(4(1(x1:S)))) -> 3(0(2(4(1(x1:S))))) 2(0(5(1(x1:S)))) -> 5(4(2(1(0(x1:S))))) 2(0(5(4(1(x1:S))))) -> 0(4(5(3(2(1(x1:S)))))) 2(2(0(1(2(x1:S))))) -> 2(4(0(2(2(1(x1:S)))))) 2(2(5(1(x1:S)))) -> 3(2(2(4(5(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 1(0(2(4(4(x1:S))))) 2(4(0(1(x1:S)))) -> 3(0(0(2(4(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 5(4(0(2(1(1(x1:S)))))) 2(4(0(5(1(x1:S))))) -> 1(4(5(0(4(2(x1:S)))))) 2(4(2(3(1(x1:S))))) -> 4(2(2(3(3(1(x1:S)))))) 2(5(2(0(1(x1:S))))) -> 0(2(4(1(5(2(x1:S)))))) 2(5(2(1(x1:S)))) -> 1(5(2(2(3(1(x1:S)))))) 2(5(1(x1:S))) -> 0(2(1(5(1(x1:S))))) 2(5(1(x1:S))) -> 1(4(5(4(2(x1:S))))) 2(5(1(x1:S))) -> 1(5(0(2(4(1(x1:S)))))) 2(5(1(x1:S))) -> 5(0(2(1(4(x1:S))))) 2(5(1(x1:S))) -> 5(2(1(1(1(1(x1:S)))))) 2(5(1(x1:S))) -> 5(2(1(4(1(x1:S))))) 2(5(4(1(x1:S)))) -> 4(5(2(1(4(4(x1:S)))))) 2(5(5(2(x1:S)))) -> 5(5(2(3(2(x1:S))))) 2(5(5(1(x1:S)))) -> 1(5(4(2(4(5(x1:S)))))) The problem is decomposed in 2 subproblems. Problem 1.1: Subterm Processor: -> Pairs: 0#(2(4(3(1(x1:S))))) -> 0#(x1:S) 0#(1(2(x1:S))) -> 0#(x1:S) 0#(5(5(2(x1:S)))) -> 0#(x1:S) -> Rules: 0(2(1(2(x1:S)))) -> 0(2(2(1(5(x1:S))))) 0(2(4(3(1(x1:S))))) -> 1(3(4(2(3(0(x1:S)))))) 0(2(4(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 0(2(5(1(x1:S)))) -> 1(1(5(0(2(1(x1:S)))))) 0(2(5(3(1(x1:S))))) -> 5(0(2(3(5(1(x1:S)))))) 0(1(2(1(x1:S)))) -> 3(1(4(0(2(1(x1:S)))))) 0(1(2(x1:S))) -> 0(2(1(0(x1:S)))) 0(1(2(x1:S))) -> 0(2(4(1(5(x1:S))))) 0(1(2(x1:S))) -> 0(1(3(4(2(3(x1:S)))))) 0(1(2(x1:S))) -> 0(3(2(1(0(x1:S))))) 0(1(2(x1:S))) -> 1(0(2(3(x1:S)))) 0(1(2(x1:S))) -> 1(0(3(2(3(x1:S))))) 0(1(3(0(1(x1:S))))) -> 0(3(1(0(1(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 1(0(3(4(2(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(1(5(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(5(2(x1:S))) -> 0(2(4(5(3(x1:S))))) 0(5(2(x1:S))) -> 5(4(2(3(0(4(x1:S)))))) 0(5(3(1(x1:S)))) -> 0(1(4(4(3(5(x1:S)))))) 0(5(5(2(x1:S)))) -> 5(4(2(3(5(0(x1:S)))))) 2(0(1(2(x1:S)))) -> 0(2(3(2(1(1(x1:S)))))) 2(0(1(2(x1:S)))) -> 4(0(2(1(1(2(x1:S)))))) 2(0(1(x1:S))) -> 0(2(2(3(4(1(x1:S)))))) 2(0(1(x1:S))) -> 0(2(1(1(4(x1:S))))) 2(0(1(x1:S))) -> 0(3(2(3(1(1(x1:S)))))) 2(0(1(x1:S))) -> 0(3(2(4(1(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(4(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(x1:S)))) 2(0(1(x1:S))) -> 4(0(4(2(1(4(x1:S)))))) 2(0(4(1(x1:S)))) -> 3(0(2(4(1(x1:S))))) 2(0(5(1(x1:S)))) -> 5(4(2(1(0(x1:S))))) 2(0(5(4(1(x1:S))))) -> 0(4(5(3(2(1(x1:S)))))) 2(2(0(1(2(x1:S))))) -> 2(4(0(2(2(1(x1:S)))))) 2(2(5(1(x1:S)))) -> 3(2(2(4(5(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 1(0(2(4(4(x1:S))))) 2(4(0(1(x1:S)))) -> 3(0(0(2(4(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 5(4(0(2(1(1(x1:S)))))) 2(4(0(5(1(x1:S))))) -> 1(4(5(0(4(2(x1:S)))))) 2(4(2(3(1(x1:S))))) -> 4(2(2(3(3(1(x1:S)))))) 2(5(2(0(1(x1:S))))) -> 0(2(4(1(5(2(x1:S)))))) 2(5(2(1(x1:S)))) -> 1(5(2(2(3(1(x1:S)))))) 2(5(1(x1:S))) -> 0(2(1(5(1(x1:S))))) 2(5(1(x1:S))) -> 1(4(5(4(2(x1:S))))) 2(5(1(x1:S))) -> 1(5(0(2(4(1(x1:S)))))) 2(5(1(x1:S))) -> 5(0(2(1(4(x1:S))))) 2(5(1(x1:S))) -> 5(2(1(1(1(1(x1:S)))))) 2(5(1(x1:S))) -> 5(2(1(4(1(x1:S))))) 2(5(4(1(x1:S)))) -> 4(5(2(1(4(4(x1:S)))))) 2(5(5(2(x1:S)))) -> 5(5(2(3(2(x1:S))))) 2(5(5(1(x1:S)))) -> 1(5(4(2(4(5(x1:S)))))) ->Projection: pi(0#) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: 0(2(1(2(x1:S)))) -> 0(2(2(1(5(x1:S))))) 0(2(4(3(1(x1:S))))) -> 1(3(4(2(3(0(x1:S)))))) 0(2(4(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 0(2(5(1(x1:S)))) -> 1(1(5(0(2(1(x1:S)))))) 0(2(5(3(1(x1:S))))) -> 5(0(2(3(5(1(x1:S)))))) 0(1(2(1(x1:S)))) -> 3(1(4(0(2(1(x1:S)))))) 0(1(2(x1:S))) -> 0(2(1(0(x1:S)))) 0(1(2(x1:S))) -> 0(2(4(1(5(x1:S))))) 0(1(2(x1:S))) -> 0(1(3(4(2(3(x1:S)))))) 0(1(2(x1:S))) -> 0(3(2(1(0(x1:S))))) 0(1(2(x1:S))) -> 1(0(2(3(x1:S)))) 0(1(2(x1:S))) -> 1(0(3(2(3(x1:S))))) 0(1(3(0(1(x1:S))))) -> 0(3(1(0(1(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 1(0(3(4(2(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(1(5(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(5(2(x1:S))) -> 0(2(4(5(3(x1:S))))) 0(5(2(x1:S))) -> 5(4(2(3(0(4(x1:S)))))) 0(5(3(1(x1:S)))) -> 0(1(4(4(3(5(x1:S)))))) 0(5(5(2(x1:S)))) -> 5(4(2(3(5(0(x1:S)))))) 2(0(1(2(x1:S)))) -> 0(2(3(2(1(1(x1:S)))))) 2(0(1(2(x1:S)))) -> 4(0(2(1(1(2(x1:S)))))) 2(0(1(x1:S))) -> 0(2(2(3(4(1(x1:S)))))) 2(0(1(x1:S))) -> 0(2(1(1(4(x1:S))))) 2(0(1(x1:S))) -> 0(3(2(3(1(1(x1:S)))))) 2(0(1(x1:S))) -> 0(3(2(4(1(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(4(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(x1:S)))) 2(0(1(x1:S))) -> 4(0(4(2(1(4(x1:S)))))) 2(0(4(1(x1:S)))) -> 3(0(2(4(1(x1:S))))) 2(0(5(1(x1:S)))) -> 5(4(2(1(0(x1:S))))) 2(0(5(4(1(x1:S))))) -> 0(4(5(3(2(1(x1:S)))))) 2(2(0(1(2(x1:S))))) -> 2(4(0(2(2(1(x1:S)))))) 2(2(5(1(x1:S)))) -> 3(2(2(4(5(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 1(0(2(4(4(x1:S))))) 2(4(0(1(x1:S)))) -> 3(0(0(2(4(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 5(4(0(2(1(1(x1:S)))))) 2(4(0(5(1(x1:S))))) -> 1(4(5(0(4(2(x1:S)))))) 2(4(2(3(1(x1:S))))) -> 4(2(2(3(3(1(x1:S)))))) 2(5(2(0(1(x1:S))))) -> 0(2(4(1(5(2(x1:S)))))) 2(5(2(1(x1:S)))) -> 1(5(2(2(3(1(x1:S)))))) 2(5(1(x1:S))) -> 0(2(1(5(1(x1:S))))) 2(5(1(x1:S))) -> 1(4(5(4(2(x1:S))))) 2(5(1(x1:S))) -> 1(5(0(2(4(1(x1:S)))))) 2(5(1(x1:S))) -> 5(0(2(1(4(x1:S))))) 2(5(1(x1:S))) -> 5(2(1(1(1(1(x1:S)))))) 2(5(1(x1:S))) -> 5(2(1(4(1(x1:S))))) 2(5(4(1(x1:S)))) -> 4(5(2(1(4(4(x1:S)))))) 2(5(5(2(x1:S)))) -> 5(5(2(3(2(x1:S))))) 2(5(5(1(x1:S)))) -> 1(5(4(2(4(5(x1:S)))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: 2#(4(0(5(1(x1:S))))) -> 2#(x1:S) 2#(5(2(0(1(x1:S))))) -> 2#(x1:S) 2#(5(1(x1:S))) -> 2#(x1:S) -> Rules: 0(2(1(2(x1:S)))) -> 0(2(2(1(5(x1:S))))) 0(2(4(3(1(x1:S))))) -> 1(3(4(2(3(0(x1:S)))))) 0(2(4(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 0(2(5(1(x1:S)))) -> 1(1(5(0(2(1(x1:S)))))) 0(2(5(3(1(x1:S))))) -> 5(0(2(3(5(1(x1:S)))))) 0(1(2(1(x1:S)))) -> 3(1(4(0(2(1(x1:S)))))) 0(1(2(x1:S))) -> 0(2(1(0(x1:S)))) 0(1(2(x1:S))) -> 0(2(4(1(5(x1:S))))) 0(1(2(x1:S))) -> 0(1(3(4(2(3(x1:S)))))) 0(1(2(x1:S))) -> 0(3(2(1(0(x1:S))))) 0(1(2(x1:S))) -> 1(0(2(3(x1:S)))) 0(1(2(x1:S))) -> 1(0(3(2(3(x1:S))))) 0(1(3(0(1(x1:S))))) -> 0(3(1(0(1(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 1(0(3(4(2(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(1(5(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(5(2(x1:S))) -> 0(2(4(5(3(x1:S))))) 0(5(2(x1:S))) -> 5(4(2(3(0(4(x1:S)))))) 0(5(3(1(x1:S)))) -> 0(1(4(4(3(5(x1:S)))))) 0(5(5(2(x1:S)))) -> 5(4(2(3(5(0(x1:S)))))) 2(0(1(2(x1:S)))) -> 0(2(3(2(1(1(x1:S)))))) 2(0(1(2(x1:S)))) -> 4(0(2(1(1(2(x1:S)))))) 2(0(1(x1:S))) -> 0(2(2(3(4(1(x1:S)))))) 2(0(1(x1:S))) -> 0(2(1(1(4(x1:S))))) 2(0(1(x1:S))) -> 0(3(2(3(1(1(x1:S)))))) 2(0(1(x1:S))) -> 0(3(2(4(1(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(4(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(x1:S)))) 2(0(1(x1:S))) -> 4(0(4(2(1(4(x1:S)))))) 2(0(4(1(x1:S)))) -> 3(0(2(4(1(x1:S))))) 2(0(5(1(x1:S)))) -> 5(4(2(1(0(x1:S))))) 2(0(5(4(1(x1:S))))) -> 0(4(5(3(2(1(x1:S)))))) 2(2(0(1(2(x1:S))))) -> 2(4(0(2(2(1(x1:S)))))) 2(2(5(1(x1:S)))) -> 3(2(2(4(5(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 1(0(2(4(4(x1:S))))) 2(4(0(1(x1:S)))) -> 3(0(0(2(4(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 5(4(0(2(1(1(x1:S)))))) 2(4(0(5(1(x1:S))))) -> 1(4(5(0(4(2(x1:S)))))) 2(4(2(3(1(x1:S))))) -> 4(2(2(3(3(1(x1:S)))))) 2(5(2(0(1(x1:S))))) -> 0(2(4(1(5(2(x1:S)))))) 2(5(2(1(x1:S)))) -> 1(5(2(2(3(1(x1:S)))))) 2(5(1(x1:S))) -> 0(2(1(5(1(x1:S))))) 2(5(1(x1:S))) -> 1(4(5(4(2(x1:S))))) 2(5(1(x1:S))) -> 1(5(0(2(4(1(x1:S)))))) 2(5(1(x1:S))) -> 5(0(2(1(4(x1:S))))) 2(5(1(x1:S))) -> 5(2(1(1(1(1(x1:S)))))) 2(5(1(x1:S))) -> 5(2(1(4(1(x1:S))))) 2(5(4(1(x1:S)))) -> 4(5(2(1(4(4(x1:S)))))) 2(5(5(2(x1:S)))) -> 5(5(2(3(2(x1:S))))) 2(5(5(1(x1:S)))) -> 1(5(4(2(4(5(x1:S)))))) ->Projection: pi(2#) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: 0(2(1(2(x1:S)))) -> 0(2(2(1(5(x1:S))))) 0(2(4(3(1(x1:S))))) -> 1(3(4(2(3(0(x1:S)))))) 0(2(4(3(1(x1:S))))) -> 4(0(3(2(1(0(x1:S)))))) 0(2(5(1(x1:S)))) -> 1(1(5(0(2(1(x1:S)))))) 0(2(5(3(1(x1:S))))) -> 5(0(2(3(5(1(x1:S)))))) 0(1(2(1(x1:S)))) -> 3(1(4(0(2(1(x1:S)))))) 0(1(2(x1:S))) -> 0(2(1(0(x1:S)))) 0(1(2(x1:S))) -> 0(2(4(1(5(x1:S))))) 0(1(2(x1:S))) -> 0(1(3(4(2(3(x1:S)))))) 0(1(2(x1:S))) -> 0(3(2(1(0(x1:S))))) 0(1(2(x1:S))) -> 1(0(2(3(x1:S)))) 0(1(2(x1:S))) -> 1(0(3(2(3(x1:S))))) 0(1(3(0(1(x1:S))))) -> 0(3(1(0(1(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 1(0(3(4(2(1(x1:S)))))) 0(1(3(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(1(5(1(x1:S)))) -> 5(0(3(1(1(x1:S))))) 0(5(2(x1:S))) -> 0(2(4(5(3(x1:S))))) 0(5(2(x1:S))) -> 5(4(2(3(0(4(x1:S)))))) 0(5(3(1(x1:S)))) -> 0(1(4(4(3(5(x1:S)))))) 0(5(5(2(x1:S)))) -> 5(4(2(3(5(0(x1:S)))))) 2(0(1(2(x1:S)))) -> 0(2(3(2(1(1(x1:S)))))) 2(0(1(2(x1:S)))) -> 4(0(2(1(1(2(x1:S)))))) 2(0(1(x1:S))) -> 0(2(2(3(4(1(x1:S)))))) 2(0(1(x1:S))) -> 0(2(1(1(4(x1:S))))) 2(0(1(x1:S))) -> 0(3(2(3(1(1(x1:S)))))) 2(0(1(x1:S))) -> 0(3(2(4(1(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(4(x1:S))))) 2(0(1(x1:S))) -> 3(0(2(1(x1:S)))) 2(0(1(x1:S))) -> 4(0(4(2(1(4(x1:S)))))) 2(0(4(1(x1:S)))) -> 3(0(2(4(1(x1:S))))) 2(0(5(1(x1:S)))) -> 5(4(2(1(0(x1:S))))) 2(0(5(4(1(x1:S))))) -> 0(4(5(3(2(1(x1:S)))))) 2(2(0(1(2(x1:S))))) -> 2(4(0(2(2(1(x1:S)))))) 2(2(5(1(x1:S)))) -> 3(2(2(4(5(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 1(0(2(4(4(x1:S))))) 2(4(0(1(x1:S)))) -> 3(0(0(2(4(1(x1:S)))))) 2(4(0(1(x1:S)))) -> 5(4(0(2(1(1(x1:S)))))) 2(4(0(5(1(x1:S))))) -> 1(4(5(0(4(2(x1:S)))))) 2(4(2(3(1(x1:S))))) -> 4(2(2(3(3(1(x1:S)))))) 2(5(2(0(1(x1:S))))) -> 0(2(4(1(5(2(x1:S)))))) 2(5(2(1(x1:S)))) -> 1(5(2(2(3(1(x1:S)))))) 2(5(1(x1:S))) -> 0(2(1(5(1(x1:S))))) 2(5(1(x1:S))) -> 1(4(5(4(2(x1:S))))) 2(5(1(x1:S))) -> 1(5(0(2(4(1(x1:S)))))) 2(5(1(x1:S))) -> 5(0(2(1(4(x1:S))))) 2(5(1(x1:S))) -> 5(2(1(1(1(1(x1:S)))))) 2(5(1(x1:S))) -> 5(2(1(4(1(x1:S))))) 2(5(4(1(x1:S)))) -> 4(5(2(1(4(4(x1:S)))))) 2(5(5(2(x1:S)))) -> 5(5(2(3(2(x1:S))))) 2(5(5(1(x1:S)))) -> 1(5(4(2(4(5(x1:S)))))) ->Strongly Connected Components: There is no strongly connected component The problem is finite.