NO Problem 1: (VAR v_NonEmpty:S x1:S) (RULES 0(q0(0(x1:S))) -> 0(0(q0(x1:S))) 0(q0(1(x1:S))) -> 0(1(q0(x1:S))) 0(q0(h(x1:S))) -> 0(0(q0(h(x1:S)))) 0(q1(0(x1:S))) -> 0(0(q2(x1:S))) 0(q1(1(x1:S))) -> 0(1(q2(x1:S))) 0(q1(h(x1:S))) -> 0(0(q2(h(x1:S)))) 0(q2(x1:S)) -> q3(1(x1:S)) 0(q3(x1:S)) -> q4(0(x1:S)) 0(q4(0(x1:S))) -> 1(0(q5(x1:S))) 0(q4(1(x1:S))) -> 1(1(q5(x1:S))) 0(q4(h(x1:S))) -> 1(0(q5(h(x1:S)))) 1(q0(0(x1:S))) -> 0(0(q1(x1:S))) 1(q0(1(x1:S))) -> 0(1(q1(x1:S))) 1(q0(h(x1:S))) -> 0(0(q1(h(x1:S)))) 1(q1(0(x1:S))) -> 1(0(q1(x1:S))) 1(q1(1(x1:S))) -> 1(1(q1(x1:S))) 1(q1(h(x1:S))) -> 1(0(q1(h(x1:S)))) 1(q2(0(x1:S))) -> 1(0(q2(x1:S))) 1(q2(1(x1:S))) -> 1(1(q2(x1:S))) 1(q2(h(x1:S))) -> 1(0(q2(h(x1:S)))) 1(q3(x1:S)) -> q3(1(x1:S)) 1(q4(x1:S)) -> q4(1(x1:S)) 1(q5(0(x1:S))) -> 0(0(q1(x1:S))) 1(q5(1(x1:S))) -> 0(1(q1(x1:S))) 1(q5(h(x1:S))) -> 0(0(q1(h(x1:S)))) h(q0(x1:S)) -> h(0(q0(x1:S))) h(q1(x1:S)) -> h(0(q1(x1:S))) h(q2(x1:S)) -> h(0(q2(x1:S))) h(q3(x1:S)) -> h(0(q3(x1:S))) h(q4(x1:S)) -> h(0(q4(x1:S))) h(q5(x1:S)) -> h(0(q5(x1:S))) ) Problem 1: Dependency Pairs Processor: -> Pairs: 0#(q0(0(x1:S))) -> 0#(0(q0(x1:S))) 0#(q0(0(x1:S))) -> 0#(q0(x1:S)) 0#(q0(1(x1:S))) -> 0#(1(q0(x1:S))) 0#(q0(1(x1:S))) -> 1#(q0(x1:S)) 0#(q0(h(x1:S))) -> 0#(0(q0(h(x1:S)))) 0#(q0(h(x1:S))) -> 0#(q0(h(x1:S))) 0#(q1(0(x1:S))) -> 0#(0(q2(x1:S))) 0#(q1(0(x1:S))) -> 0#(q2(x1:S)) 0#(q1(1(x1:S))) -> 0#(1(q2(x1:S))) 0#(q1(1(x1:S))) -> 1#(q2(x1:S)) 0#(q1(h(x1:S))) -> 0#(0(q2(h(x1:S)))) 0#(q1(h(x1:S))) -> 0#(q2(h(x1:S))) 0#(q2(x1:S)) -> 1#(x1:S) 0#(q3(x1:S)) -> 0#(x1:S) 0#(q4(1(x1:S))) -> 1#(1(q5(x1:S))) 0#(q4(1(x1:S))) -> 1#(q5(x1:S)) 0#(q4(h(x1:S))) -> 0#(q5(h(x1:S))) 0#(q4(h(x1:S))) -> 1#(0(q5(h(x1:S)))) 1#(q0(0(x1:S))) -> 0#(0(q1(x1:S))) 1#(q0(0(x1:S))) -> 0#(q1(x1:S)) 1#(q0(1(x1:S))) -> 0#(1(q1(x1:S))) 1#(q0(1(x1:S))) -> 1#(q1(x1:S)) 1#(q0(h(x1:S))) -> 0#(0(q1(h(x1:S)))) 1#(q0(h(x1:S))) -> 0#(q1(h(x1:S))) 1#(q1(0(x1:S))) -> 0#(q1(x1:S)) 1#(q1(0(x1:S))) -> 1#(0(q1(x1:S))) 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S))) 1#(q1(1(x1:S))) -> 1#(q1(x1:S)) 1#(q1(h(x1:S))) -> 0#(q1(h(x1:S))) 1#(q1(h(x1:S))) -> 1#(0(q1(h(x1:S)))) 1#(q2(0(x1:S))) -> 0#(q2(x1:S)) 1#(q2(0(x1:S))) -> 1#(0(q2(x1:S))) 1#(q2(1(x1:S))) -> 1#(1(q2(x1:S))) 1#(q2(1(x1:S))) -> 1#(q2(x1:S)) 1#(q2(h(x1:S))) -> 0#(q2(h(x1:S))) 1#(q2(h(x1:S))) -> 1#(0(q2(h(x1:S)))) 1#(q3(x1:S)) -> 1#(x1:S) 1#(q4(x1:S)) -> 1#(x1:S) 1#(q5(0(x1:S))) -> 0#(0(q1(x1:S))) 1#(q5(0(x1:S))) -> 0#(q1(x1:S)) 1#(q5(1(x1:S))) -> 0#(1(q1(x1:S))) 1#(q5(1(x1:S))) -> 1#(q1(x1:S)) 1#(q5(h(x1:S))) -> 0#(0(q1(h(x1:S)))) 1#(q5(h(x1:S))) -> 0#(q1(h(x1:S))) H(q0(x1:S)) -> 0#(q0(x1:S)) H(q0(x1:S)) -> H(0(q0(x1:S))) H(q1(x1:S)) -> 0#(q1(x1:S)) H(q1(x1:S)) -> H(0(q1(x1:S))) H(q2(x1:S)) -> 0#(q2(x1:S)) H(q2(x1:S)) -> H(0(q2(x1:S))) H(q3(x1:S)) -> 0#(q3(x1:S)) H(q3(x1:S)) -> H(0(q3(x1:S))) H(q4(x1:S)) -> 0#(q4(x1:S)) H(q4(x1:S)) -> H(0(q4(x1:S))) -> Rules: 0(q0(0(x1:S))) -> 0(0(q0(x1:S))) 0(q0(1(x1:S))) -> 0(1(q0(x1:S))) 0(q0(h(x1:S))) -> 0(0(q0(h(x1:S)))) 0(q1(0(x1:S))) -> 0(0(q2(x1:S))) 0(q1(1(x1:S))) -> 0(1(q2(x1:S))) 0(q1(h(x1:S))) -> 0(0(q2(h(x1:S)))) 0(q2(x1:S)) -> q3(1(x1:S)) 0(q3(x1:S)) -> q4(0(x1:S)) 0(q4(0(x1:S))) -> 1(0(q5(x1:S))) 0(q4(1(x1:S))) -> 1(1(q5(x1:S))) 0(q4(h(x1:S))) -> 1(0(q5(h(x1:S)))) 1(q0(0(x1:S))) -> 0(0(q1(x1:S))) 1(q0(1(x1:S))) -> 0(1(q1(x1:S))) 1(q0(h(x1:S))) -> 0(0(q1(h(x1:S)))) 1(q1(0(x1:S))) -> 1(0(q1(x1:S))) 1(q1(1(x1:S))) -> 1(1(q1(x1:S))) 1(q1(h(x1:S))) -> 1(0(q1(h(x1:S)))) 1(q2(0(x1:S))) -> 1(0(q2(x1:S))) 1(q2(1(x1:S))) -> 1(1(q2(x1:S))) 1(q2(h(x1:S))) -> 1(0(q2(h(x1:S)))) 1(q3(x1:S)) -> q3(1(x1:S)) 1(q4(x1:S)) -> q4(1(x1:S)) 1(q5(0(x1:S))) -> 0(0(q1(x1:S))) 1(q5(1(x1:S))) -> 0(1(q1(x1:S))) 1(q5(h(x1:S))) -> 0(0(q1(h(x1:S)))) h(q0(x1:S)) -> h(0(q0(x1:S))) h(q1(x1:S)) -> h(0(q1(x1:S))) h(q2(x1:S)) -> h(0(q2(x1:S))) h(q3(x1:S)) -> h(0(q3(x1:S))) h(q4(x1:S)) -> h(0(q4(x1:S))) h(q5(x1:S)) -> h(0(q5(x1:S))) Problem 1: Infinite Processor: -> Pairs: 0#(q0(0(x1:S))) -> 0#(0(q0(x1:S))) 0#(q0(0(x1:S))) -> 0#(q0(x1:S)) 0#(q0(1(x1:S))) -> 0#(1(q0(x1:S))) 0#(q0(1(x1:S))) -> 1#(q0(x1:S)) 0#(q0(h(x1:S))) -> 0#(0(q0(h(x1:S)))) 0#(q0(h(x1:S))) -> 0#(q0(h(x1:S))) 0#(q1(0(x1:S))) -> 0#(0(q2(x1:S))) 0#(q1(0(x1:S))) -> 0#(q2(x1:S)) 0#(q1(1(x1:S))) -> 0#(1(q2(x1:S))) 0#(q1(1(x1:S))) -> 1#(q2(x1:S)) 0#(q1(h(x1:S))) -> 0#(0(q2(h(x1:S)))) 0#(q1(h(x1:S))) -> 0#(q2(h(x1:S))) 0#(q2(x1:S)) -> 1#(x1:S) 0#(q3(x1:S)) -> 0#(x1:S) 0#(q4(1(x1:S))) -> 1#(1(q5(x1:S))) 0#(q4(1(x1:S))) -> 1#(q5(x1:S)) 0#(q4(h(x1:S))) -> 0#(q5(h(x1:S))) 0#(q4(h(x1:S))) -> 1#(0(q5(h(x1:S)))) 1#(q0(0(x1:S))) -> 0#(0(q1(x1:S))) 1#(q0(0(x1:S))) -> 0#(q1(x1:S)) 1#(q0(1(x1:S))) -> 0#(1(q1(x1:S))) 1#(q0(1(x1:S))) -> 1#(q1(x1:S)) 1#(q0(h(x1:S))) -> 0#(0(q1(h(x1:S)))) 1#(q0(h(x1:S))) -> 0#(q1(h(x1:S))) 1#(q1(0(x1:S))) -> 0#(q1(x1:S)) 1#(q1(0(x1:S))) -> 1#(0(q1(x1:S))) 1#(q1(1(x1:S))) -> 1#(1(q1(x1:S))) 1#(q1(1(x1:S))) -> 1#(q1(x1:S)) 1#(q1(h(x1:S))) -> 0#(q1(h(x1:S))) 1#(q1(h(x1:S))) -> 1#(0(q1(h(x1:S)))) 1#(q2(0(x1:S))) -> 0#(q2(x1:S)) 1#(q2(0(x1:S))) -> 1#(0(q2(x1:S))) 1#(q2(1(x1:S))) -> 1#(1(q2(x1:S))) 1#(q2(1(x1:S))) -> 1#(q2(x1:S)) 1#(q2(h(x1:S))) -> 0#(q2(h(x1:S))) 1#(q2(h(x1:S))) -> 1#(0(q2(h(x1:S)))) 1#(q3(x1:S)) -> 1#(x1:S) 1#(q4(x1:S)) -> 1#(x1:S) 1#(q5(0(x1:S))) -> 0#(0(q1(x1:S))) 1#(q5(0(x1:S))) -> 0#(q1(x1:S)) 1#(q5(1(x1:S))) -> 0#(1(q1(x1:S))) 1#(q5(1(x1:S))) -> 1#(q1(x1:S)) 1#(q5(h(x1:S))) -> 0#(0(q1(h(x1:S)))) 1#(q5(h(x1:S))) -> 0#(q1(h(x1:S))) H(q0(x1:S)) -> 0#(q0(x1:S)) H(q0(x1:S)) -> H(0(q0(x1:S))) H(q1(x1:S)) -> 0#(q1(x1:S)) H(q1(x1:S)) -> H(0(q1(x1:S))) H(q2(x1:S)) -> 0#(q2(x1:S)) H(q2(x1:S)) -> H(0(q2(x1:S))) H(q3(x1:S)) -> 0#(q3(x1:S)) H(q3(x1:S)) -> H(0(q3(x1:S))) H(q4(x1:S)) -> 0#(q4(x1:S)) H(q4(x1:S)) -> H(0(q4(x1:S))) -> Rules: 0(q0(0(x1:S))) -> 0(0(q0(x1:S))) 0(q0(1(x1:S))) -> 0(1(q0(x1:S))) 0(q0(h(x1:S))) -> 0(0(q0(h(x1:S)))) 0(q1(0(x1:S))) -> 0(0(q2(x1:S))) 0(q1(1(x1:S))) -> 0(1(q2(x1:S))) 0(q1(h(x1:S))) -> 0(0(q2(h(x1:S)))) 0(q2(x1:S)) -> q3(1(x1:S)) 0(q3(x1:S)) -> q4(0(x1:S)) 0(q4(0(x1:S))) -> 1(0(q5(x1:S))) 0(q4(1(x1:S))) -> 1(1(q5(x1:S))) 0(q4(h(x1:S))) -> 1(0(q5(h(x1:S)))) 1(q0(0(x1:S))) -> 0(0(q1(x1:S))) 1(q0(1(x1:S))) -> 0(1(q1(x1:S))) 1(q0(h(x1:S))) -> 0(0(q1(h(x1:S)))) 1(q1(0(x1:S))) -> 1(0(q1(x1:S))) 1(q1(1(x1:S))) -> 1(1(q1(x1:S))) 1(q1(h(x1:S))) -> 1(0(q1(h(x1:S)))) 1(q2(0(x1:S))) -> 1(0(q2(x1:S))) 1(q2(1(x1:S))) -> 1(1(q2(x1:S))) 1(q2(h(x1:S))) -> 1(0(q2(h(x1:S)))) 1(q3(x1:S)) -> q3(1(x1:S)) 1(q4(x1:S)) -> q4(1(x1:S)) 1(q5(0(x1:S))) -> 0(0(q1(x1:S))) 1(q5(1(x1:S))) -> 0(1(q1(x1:S))) 1(q5(h(x1:S))) -> 0(0(q1(h(x1:S)))) h(q0(x1:S)) -> h(0(q0(x1:S))) h(q1(x1:S)) -> h(0(q1(x1:S))) h(q2(x1:S)) -> h(0(q2(x1:S))) h(q3(x1:S)) -> h(0(q3(x1:S))) h(q4(x1:S)) -> h(0(q4(x1:S))) h(q5(x1:S)) -> h(0(q5(x1:S))) -> Pairs in cycle: 0#(q0(h(x1:S))) -> 0#(q0(h(x1:S))) The problem is infinite.