Input TRS: 1: active(from(X)) -> mark(cons(X,from(s(X)))) 2: active(sel(0(),cons(X,Y))) -> mark(X) 3: active(sel(s(X),cons(Y,Z))) -> mark(sel(X,Z)) 4: active(from(X)) -> from(active(X)) 5: active(cons(X1,X2)) -> cons(active(X1),X2) 6: active(s(X)) -> s(active(X)) 7: active(sel(X1,X2)) -> sel(active(X1),X2) 8: active(sel(X1,X2)) -> sel(X1,active(X2)) 9: from(mark(X)) -> mark(from(X)) 10: cons(mark(X1),X2) -> mark(cons(X1,X2)) 11: s(mark(X)) -> mark(s(X)) 12: sel(mark(X1),X2) -> mark(sel(X1,X2)) 13: sel(X1,mark(X2)) -> mark(sel(X1,X2)) 14: proper(from(X)) -> from(proper(X)) 15: proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) 16: proper(s(X)) -> s(proper(X)) 17: proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) 18: proper(0()) -> ok(0()) 19: from(ok(X)) -> ok(from(X)) 20: cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) 21: s(ok(X)) -> ok(s(X)) 22: sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) 23: top(mark(X)) -> top(proper(X)) 24: top(ok(X)) -> top(active(X)) Number of strict rules: 24 Direct Order(PosReal,>,Poly) ... failed. Freezing ... failed. Dependency Pairs: #1: #active(s(X)) -> #s(active(X)) #2: #active(s(X)) -> #active(X) #3: #sel(X1,mark(X2)) -> #sel(X1,X2) #4: #from(mark(X)) -> #from(X) #5: #s(mark(X)) -> #s(X) #6: #top(ok(X)) -> #top(active(X)) #7: #top(ok(X)) -> #active(X) #8: #top(mark(X)) -> #top(proper(X)) #9: #top(mark(X)) -> #proper(X) #10: #sel(mark(X1),X2) -> #sel(X1,X2) #11: #proper(from(X)) -> #from(proper(X)) #12: #proper(from(X)) -> #proper(X) #13: #cons(ok(X1),ok(X2)) -> #cons(X1,X2) #14: #active(sel(X1,X2)) -> #sel(active(X1),X2) #15: #active(sel(X1,X2)) -> #active(X1) #16: #cons(mark(X1),X2) -> #cons(X1,X2) #17: #active(cons(X1,X2)) -> #cons(active(X1),X2) #18: #active(cons(X1,X2)) -> #active(X1) #19: #sel(ok(X1),ok(X2)) -> #sel(X1,X2) #20: #proper(sel(X1,X2)) -> #sel(proper(X1),proper(X2)) #21: #proper(sel(X1,X2)) -> #proper(X1) #22: #proper(sel(X1,X2)) -> #proper(X2) #23: #from(ok(X)) -> #from(X) #24: #s(ok(X)) -> #s(X) #25: #proper(s(X)) -> #s(proper(X)) #26: #proper(s(X)) -> #proper(X) #27: #active(sel(s(X),cons(Y,Z))) -> #sel(X,Z) #28: #active(from(X)) -> #cons(X,from(s(X))) #29: #active(from(X)) -> #from(s(X)) #30: #active(from(X)) -> #s(X) #31: #active(sel(X1,X2)) -> #sel(X1,active(X2)) #32: #active(sel(X1,X2)) -> #active(X2) #33: #proper(cons(X1,X2)) -> #cons(proper(X1),proper(X2)) #34: #proper(cons(X1,X2)) -> #proper(X1) #35: #proper(cons(X1,X2)) -> #proper(X2) #36: #active(from(X)) -> #from(active(X)) #37: #active(from(X)) -> #active(X) Number of SCCs: 7, DPs: 22, edges: 86 SCC { #4 #23 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 top(x1) weight: 0 #top(x1) weight: 0 proper(x1) weight: 0 ok(x1) weight: (/ 1 2) + x1 0() weight: 0 #sel(x1,x2) weight: 0 from(x1) weight: 0 sel(x1,x2) weight: 0 #s(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #proper(x1) weight: 0 #from(x1) weight: x1 active(x1) weight: 0 cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { } Removed DPs: #4 #23 Number of SCCs: 6, DPs: 20, edges: 82 SCC { #5 #24 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: 0 top(x1) weight: 0 #top(x1) weight: 0 proper(x1) weight: 0 ok(x1) weight: (/ 1 2) + x1 0() weight: 0 #sel(x1,x2) weight: 0 from(x1) weight: 0 sel(x1,x2) weight: 0 #s(x1) weight: x1 mark(x1) weight: (/ 1 2) + x1 #proper(x1) weight: 0 #from(x1) weight: 0 active(x1) weight: 0 cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { } Removed DPs: #5 #24 Number of SCCs: 5, DPs: 18, edges: 78 SCC { #6 #8 } Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded. #cons(x1,x2) weight: (/ 1 8) + x2 + x1 status: [x1,x2] precedence above: s(x1) weight: x1 status: [x1] precedence above: mark top(x1) weight: (/ 1 8) status: [] precedence above: #top(x1) weight: (/ 1 8) + x1 status: [x1] precedence above: proper(x1) weight: x1 status: x1 ok(x1) weight: x1 status: x1 0() weight: (/ 1 8) status: [] precedence above: #sel(x1,x2) weight: (/ 1 8) + x2 status: [x2] precedence above: from(x1) weight: (/ 1 4) + x1 status: [x1] precedence above: s mark cons sel(x1,x2) weight: (/ 1 4) + x2 + x1 status: [x1,x2] precedence above: s mark #s(x1) weight: x1 status: [] precedence above: mark(x1) weight: x1 status: [x1] precedence above: s #proper(x1) weight: (/ 1 8) status: [] precedence above: #from(x1) weight: (/ 1 8) status: [] precedence above: active(x1) weight: x1 status: x1 cons(x1,x2) weight: max{x2, (/ 1 8) + x1} status: [x1] precedence above: s mark #active(x1) weight: x1 status: [] precedence above: Usable rules: { 1..22 } Removed DPs: #8 Number of SCCs: 5, DPs: 17, edges: 75 SCC { #6 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: (/ 1 4) + x1 top(x1) weight: 0 #top(x1) weight: x1 proper(x1) weight: (/ 1 2) + x1 ok(x1) weight: (/ 1 2) + x1 0() weight: 0 #sel(x1,x2) weight: 0 from(x1) weight: (/ 1 4) + x1 sel(x1,x2) weight: (/ 1 4) + x1 #s(x1) weight: 0 mark(x1) weight: (/ 1 4) #proper(x1) weight: 0 #from(x1) weight: 0 active(x1) weight: (/ 1 4) + x1 cons(x1,x2) weight: (/ 1 4) + x1 #active(x1) weight: 0 Usable rules: { 1..13 19..22 } Removed DPs: #6 Number of SCCs: 4, DPs: 16, edges: 74 SCC { #13 #16 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: x2 s(x1) weight: (/ 1 4) + x1 top(x1) weight: 0 #top(x1) weight: x1 proper(x1) weight: (/ 1 4) + x1 ok(x1) weight: (/ 1 4) + x1 0() weight: 0 #sel(x1,x2) weight: 0 from(x1) weight: (/ 1 4) + x1 sel(x1,x2) weight: (/ 1 4) + x1 #s(x1) weight: 0 mark(x1) weight: (/ 1 4) #proper(x1) weight: 0 #from(x1) weight: 0 active(x1) weight: (/ 1 4) + x1 cons(x1,x2) weight: (/ 1 4) + x1 #active(x1) weight: 0 Usable rules: { } Removed DPs: #13 Number of SCCs: 5, DPs: 15, edges: 71 SCC { #16 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: x1 s(x1) weight: (/ 1 4) top(x1) weight: 0 #top(x1) weight: 0 proper(x1) weight: (/ 1 4) + x1 ok(x1) weight: (/ 1 4) + x1 0() weight: 0 #sel(x1,x2) weight: 0 from(x1) weight: (/ 1 4) + x1 sel(x1,x2) weight: (/ 1 4) + x1 #s(x1) weight: 0 mark(x1) weight: (/ 1 4) + x1 #proper(x1) weight: 0 #from(x1) weight: 0 active(x1) weight: (/ 1 4) + x1 cons(x1,x2) weight: (/ 1 4) + x1 #active(x1) weight: 0 Usable rules: { } Removed DPs: #16 Number of SCCs: 4, DPs: 14, edges: 70 SCC { #3 #10 #19 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: (/ 1 4) top(x1) weight: 0 #top(x1) weight: 0 proper(x1) weight: (/ 1 4) + x1 ok(x1) weight: (/ 1 4) + x1 0() weight: 0 #sel(x1,x2) weight: x2 from(x1) weight: (/ 1 4) + x1 sel(x1,x2) weight: (/ 1 4) + x1 #s(x1) weight: 0 mark(x1) weight: (/ 1 4) + x1 #proper(x1) weight: 0 #from(x1) weight: 0 active(x1) weight: (/ 1 4) + x1 cons(x1,x2) weight: (/ 1 4) + x1 #active(x1) weight: 0 Usable rules: { } Removed DPs: #3 #19 Number of SCCs: 5, DPs: 12, edges: 62 SCC { #10 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: (/ 1 4) top(x1) weight: 0 #top(x1) weight: 0 proper(x1) weight: (/ 1 4) + x1 ok(x1) weight: (/ 1 4) + x1 0() weight: 0 #sel(x1,x2) weight: x1 from(x1) weight: (/ 1 4) + x1 sel(x1,x2) weight: (/ 1 4) + x1 #s(x1) weight: 0 mark(x1) weight: (/ 1 4) + x1 #proper(x1) weight: 0 #from(x1) weight: 0 active(x1) weight: (/ 1 4) + x1 cons(x1,x2) weight: (/ 1 4) + x1 #active(x1) weight: 0 Usable rules: { } Removed DPs: #10 Number of SCCs: 4, DPs: 11, edges: 61 SCC { #2 #15 #18 #32 #37 } 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 proper(x1) weight: (/ 1 4) + x1 ok(x1) weight: (/ 1 4) + x1 0() weight: 0 #sel(x1,x2) weight: 0 from(x1) weight: (/ 1 4) + x1 sel(x1,x2) weight: (/ 1 4) + x1 + x2 #s(x1) weight: 0 mark(x1) weight: (/ 1 4) + x1 #proper(x1) weight: 0 #from(x1) weight: 0 active(x1) weight: (/ 1 4) + x1 cons(x1,x2) weight: (/ 1 4) + x1 #active(x1) weight: x1 Usable rules: { } Removed DPs: #2 #15 #18 #32 #37 Number of SCCs: 4, DPs: 6, edges: 36 SCC { #12 #21 #22 #26 #34 #35 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #cons(x1,x2) weight: 0 s(x1) weight: (/ 1 8) + x1 top(x1) weight: 0 #top(x1) weight: 0 proper(x1) weight: (/ 1 8) + x1 ok(x1) weight: (/ 1 8) + x1 0() weight: 0 #sel(x1,x2) weight: 0 from(x1) weight: (/ 1 8) + x1 sel(x1,x2) weight: (/ 1 8) + x1 + x2 #s(x1) weight: 0 mark(x1) weight: (/ 1 8) + x1 #proper(x1) weight: x1 #from(x1) weight: 0 active(x1) weight: (/ 3 8) + x1 cons(x1,x2) weight: (/ 1 8) + x1 + x2 #active(x1) weight: 0 Usable rules: { } Removed DPs: #12 #21 #22 #26 #34 #35 Number of SCCs: 3, DPs: 0, edges: 0 YES