Input TRS: 1: active(f(0())) -> mark(cons(0(),f(s(0())))) 2: active(f(s(0()))) -> mark(f(p(s(0())))) 3: active(p(s(0()))) -> mark(0()) 4: active(f(X)) -> f(active(X)) 5: active(cons(X1,X2)) -> cons(active(X1),X2) 6: active(s(X)) -> s(active(X)) 7: active(p(X)) -> p(active(X)) 8: f(mark(X)) -> mark(f(X)) 9: cons(mark(X1),X2) -> mark(cons(X1,X2)) 10: s(mark(X)) -> mark(s(X)) 11: p(mark(X)) -> mark(p(X)) 12: proper(f(X)) -> f(proper(X)) 13: proper(0()) -> ok(0()) 14: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) 15: proper(s(X)) -> s(proper(X)) 16: proper(p(X)) -> p(proper(X)) 17: f(ok(X)) -> ok(f(X)) 18: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) 19: s(ok(X)) -> ok(s(X)) 20: p(ok(X)) -> ok(p(X)) 21: top(mark(X)) -> top(proper(X)) 22: top(ok(X)) -> top(active(X)) Number of strict rules: 22 Direct Order(PosReal,>,Poly) ... failed. Freezing ... failed. Dependency Pairs: #1: #active(f(s(0()))) -> #f(p(s(0()))) #2: #active(f(s(0()))) -> #p(s(0())) #3: #active(s(X)) -> #s(active(X)) #4: #active(s(X)) -> #active(X) #5: #cons(mark(X1),X2) -> #cons(X1,X2) #6: #p(mark(X)) -> #p(X) #7: #proper(f(X)) -> #f(proper(X)) #8: #proper(f(X)) -> #proper(X) #9: #proper(cons(X1,X2)) -> #cons(proper(X1),proper(X2)) #10: #proper(cons(X1,X2)) -> #proper(X1) #11: #proper(cons(X1,X2)) -> #proper(X2) #12: #p(ok(X)) -> #p(X) #13: #active(p(X)) -> #p(active(X)) #14: #active(p(X)) -> #active(X) #15: #s(mark(X)) -> #s(X) #16: #active(cons(X1,X2)) -> #cons(active(X1),X2) #17: #active(cons(X1,X2)) -> #active(X1) #18: #top(ok(X)) -> #top(active(X)) #19: #top(ok(X)) -> #active(X) #20: #f(ok(X)) -> #f(X) #21: #s(ok(X)) -> #s(X) #22: #top(mark(X)) -> #top(proper(X)) #23: #top(mark(X)) -> #proper(X) #24: #proper(p(X)) -> #p(proper(X)) #25: #proper(p(X)) -> #proper(X) #26: #active(f(0())) -> #cons(0(),f(s(0()))) #27: #active(f(0())) -> #f(s(0())) #28: #active(f(0())) -> #s(0()) #29: #f(mark(X)) -> #f(X) #30: #proper(s(X)) -> #s(proper(X)) #31: #proper(s(X)) -> #proper(X) #32: #active(f(X)) -> #f(active(X)) #33: #active(f(X)) -> #active(X) #34: #cons(ok(X1),ok(X2)) -> #cons(X1,X2) Number of SCCs: 7, DPs: 19, edges: 61 SCC { #20 #29 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 top(x1) weight: 0 #top(x1) weight: 0 #p(x1) weight: 0 f(x1) weight: 0 p(x1) weight: 0 proper(x1) weight: 0 ok(x1) weight: (/ 1 2) + x1 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #f(x1) weight: x1 #proper(x1) weight: 0 active(x1) weight: 0 cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { } Removed DPs: #20 #29 Number of SCCs: 6, DPs: 17, edges: 57 SCC { #15 #21 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 top(x1) weight: 0 #top(x1) weight: 0 #p(x1) weight: 0 f(x1) weight: 0 p(x1) weight: 0 proper(x1) weight: 0 ok(x1) weight: (/ 1 2) + x1 0() weight: 0 #s(x1) weight: x1 mark(x1) weight: (/ 1 2) + x1 #f(x1) weight: 0 #proper(x1) weight: 0 active(x1) weight: 0 cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { } Removed DPs: #15 #21 Number of SCCs: 5, DPs: 15, edges: 53 SCC { #6 #12 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 top(x1) weight: 0 #top(x1) weight: 0 #p(x1) weight: x1 f(x1) weight: 0 p(x1) weight: 0 proper(x1) weight: 0 ok(x1) weight: (/ 1 2) + x1 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #f(x1) weight: 0 #proper(x1) weight: 0 active(x1) weight: 0 cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { } Removed DPs: #6 #12 Number of SCCs: 4, DPs: 13, edges: 49 SCC { #18 #22 } Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)... succeeded. #cons(x1,x2) weight: 0; 0 s(x1) weight: x1_1; (/ 1 4) + x1_2 top(x1) weight: 0; 0 #top(x1) weight: x1_1; 0 #p(x1) weight: 0; 0 f(x1) weight: (/ 3 16) + x1_2; (/ 3 16) + x1_2 p(x1) weight: (/ 1 8) + x1_1; (/ 1 8) + x1_1 proper(x1) weight: (/ 1 8) + x1_1; (/ 1 8) + x1_2 ok(x1) weight: (/ 1 8) + x1_1; (/ 1 8) + x1_2 0() weight: (/ 1 16); (/ 1 16) #s(x1) weight: 0; 0 mark(x1) weight: (/ 3 16) + x1_1; (/ 3 16) + x1_2 #f(x1) weight: 0; 0 #proper(x1) weight: 0; 0 active(x1) weight: (/ 1 16) + x1_1; (/ 1 16) + x1_2 cons(x1,x2) weight: (/ 1 16) + x1_1; (/ 1 16) + x1_2 #active(x1) weight: 0; 0 Usable rules: { 1..20 } Removed DPs: #18 #22 Number of SCCs: 3, DPs: 11, edges: 45 SCC { #5 #34 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: x2 s(x1) weight: (/ 1 4) + x1 top(x1) weight: 0 #top(x1) weight: 0 #p(x1) weight: 0 f(x1) weight: (/ 1 4) + x1 p(x1) weight: (/ 1 2) proper(x1) weight: (/ 1 4) ok(x1) weight: (/ 1 2) + x1 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 3 4) #f(x1) weight: 0 #proper(x1) weight: 0 active(x1) weight: (/ 1 4) cons(x1,x2) weight: (/ 1 2) #active(x1) weight: 0 Usable rules: { } Removed DPs: #34 Number of SCCs: 3, DPs: 10, edges: 42 SCC { #5 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: x1 s(x1) weight: (/ 1 4) + x1 top(x1) weight: 0 #top(x1) weight: 0 #p(x1) weight: 0 f(x1) weight: x1 p(x1) weight: 0 proper(x1) weight: (/ 1 4) ok(x1) weight: (/ 1 2) + x1 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #f(x1) weight: 0 #proper(x1) weight: 0 active(x1) weight: (/ 1 4) cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { } Removed DPs: #5 Number of SCCs: 2, DPs: 9, edges: 41 SCC { #4 #14 #17 #33 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: (/ 1 4) + x1 top(x1) weight: 0 #top(x1) weight: 0 #p(x1) weight: 0 f(x1) weight: x1 p(x1) weight: (/ 1 4) + x1 proper(x1) weight: (/ 1 4) ok(x1) weight: (/ 1 2) + x1 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #f(x1) weight: 0 #proper(x1) weight: 0 active(x1) weight: (/ 1 4) cons(x1,x2) weight: (/ 1 4) + x1 #active(x1) weight: x1 Usable rules: { } Removed DPs: #4 #14 #17 Number of SCCs: 3, DPs: 6, edges: 26 SCC { #33 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: (/ 1 4) + x1 top(x1) weight: 0 #top(x1) weight: 0 #p(x1) weight: 0 f(x1) weight: (/ 1 4) + x1 p(x1) weight: (/ 1 4) + x1 proper(x1) weight: (/ 1 4) ok(x1) weight: (/ 1 2) + x1 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #f(x1) weight: 0 #proper(x1) weight: 0 active(x1) weight: (/ 1 4) cons(x1,x2) weight: (/ 1 4) + x1 #active(x1) weight: x1 Usable rules: { } Removed DPs: #33 Number of SCCs: 2, DPs: 5, edges: 25 SCC { #8 #10 #11 #25 #31 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: (/ 1 4) + x1 top(x1) weight: 0 #top(x1) weight: 0 #p(x1) weight: 0 f(x1) weight: (/ 1 4) + x1 p(x1) weight: (/ 1 4) + x1 proper(x1) weight: (/ 1 4) ok(x1) weight: (/ 1 2) + x1 0() weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #f(x1) weight: 0 #proper(x1) weight: x1 active(x1) weight: (/ 1 4) cons(x1,x2) weight: (/ 1 4) + x1 + x2 #active(x1) weight: 0 Usable rules: { } Removed DPs: #8 #10 #11 #25 #31 Number of SCCs: 2, DPs: 0, edges: 0 YES