Input TRS: 1: active(nats()) -> mark(adx(zeros())) 2: active(zeros()) -> mark(cons(0(),zeros())) 3: active(incr(cons(X,Y))) -> mark(cons(s(X),incr(Y))) 4: active(adx(cons(X,Y))) -> mark(incr(cons(X,adx(Y)))) 5: active(hd(cons(X,Y))) -> mark(X) 6: active(tl(cons(X,Y))) -> mark(Y) 7: mark(nats()) -> active(nats()) 8: mark(adx(X)) -> active(adx(mark(X))) 9: mark(zeros()) -> active(zeros()) 10: mark(cons(X1,X2)) -> active(cons(X1,X2)) 11: mark(0()) -> active(0()) 12: mark(incr(X)) -> active(incr(mark(X))) 13: mark(s(X)) -> active(s(X)) 14: mark(hd(X)) -> active(hd(mark(X))) 15: mark(tl(X)) -> active(tl(mark(X))) 16: adx(mark(X)) -> adx(X) 17: adx(active(X)) -> adx(X) 18: cons(mark(X1),X2) -> cons(X1,X2) 19: cons(X1,mark(X2)) -> cons(X1,X2) 20: cons(active(X1),X2) -> cons(X1,X2) 21: cons(X1,active(X2)) -> cons(X1,X2) 22: incr(mark(X)) -> incr(X) 23: incr(active(X)) -> incr(X) 24: s(mark(X)) -> s(X) 25: s(active(X)) -> s(X) 26: hd(mark(X)) -> hd(X) 27: hd(active(X)) -> hd(X) 28: tl(mark(X)) -> tl(X) 29: tl(active(X)) -> tl(X) Number of strict rules: 29 Direct Order(PosReal,>,Poly) ... removes: 5 6 incr(x1) weight: x1 hd(x1) weight: 17888 + 2 * x1 s(x1) weight: x1 adx(x1) weight: x1 zeros() weight: 0 0() weight: 0 tl(x1) weight: (/ 3 2) + x1 mark(x1) weight: x1 nats() weight: 0 active(x1) weight: x1 cons(x1,x2) weight: x1 + x2 Number of strict rules: 27 Direct Order(PosReal,>,Poly) ... failed. Freezing ... failed. Dependency Pairs: #1: #active(zeros()) -> #mark(cons(0(),zeros())) #2: #active(zeros()) -> #cons(0(),zeros()) #3: #tl(active(X)) -> #tl(X) #4: #mark(s(X)) -> #active(s(X)) #5: #mark(zeros()) -> #active(zeros()) #6: #mark(0()) -> #active(0()) #7: #s(mark(X)) -> #s(X) #8: #incr(active(X)) -> #incr(X) #9: #mark(incr(X)) -> #active(incr(mark(X))) #10: #mark(incr(X)) -> #incr(mark(X)) #11: #mark(incr(X)) -> #mark(X) #12: #mark(hd(X)) -> #active(hd(mark(X))) #13: #mark(hd(X)) -> #hd(mark(X)) #14: #mark(hd(X)) -> #mark(X) #15: #s(active(X)) -> #s(X) #16: #cons(active(X1),X2) -> #cons(X1,X2) #17: #mark(nats()) -> #active(nats()) #18: #mark(cons(X1,X2)) -> #active(cons(X1,X2)) #19: #tl(mark(X)) -> #tl(X) #20: #incr(mark(X)) -> #incr(X) #21: #hd(active(X)) -> #hd(X) #22: #adx(active(X)) -> #adx(X) #23: #cons(X1,mark(X2)) -> #cons(X1,X2) #24: #hd(mark(X)) -> #hd(X) #25: #cons(X1,active(X2)) -> #cons(X1,X2) #26: #adx(mark(X)) -> #adx(X) #27: #active(incr(cons(X,Y))) -> #mark(cons(s(X),incr(Y))) #28: #active(incr(cons(X,Y))) -> #cons(s(X),incr(Y)) #29: #active(incr(cons(X,Y))) -> #s(X) #30: #active(incr(cons(X,Y))) -> #incr(Y) #31: #active(nats()) -> #mark(adx(zeros())) #32: #active(nats()) -> #adx(zeros()) #33: #mark(adx(X)) -> #active(adx(mark(X))) #34: #mark(adx(X)) -> #adx(mark(X)) #35: #mark(adx(X)) -> #mark(X) #36: #mark(tl(X)) -> #active(tl(mark(X))) #37: #mark(tl(X)) -> #tl(mark(X)) #38: #mark(tl(X)) -> #mark(X) #39: #active(adx(cons(X,Y))) -> #mark(incr(cons(X,adx(Y)))) #40: #active(adx(cons(X,Y))) -> #incr(cons(X,adx(Y))) #41: #active(adx(cons(X,Y))) -> #cons(X,adx(Y)) #42: #active(adx(cons(X,Y))) -> #adx(Y) #43: #cons(mark(X1),X2) -> #cons(X1,X2) Number of SCCs: 7, DPs: 22, edges: 65 SCC { #8 #20 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #adx(x1) weight: 0 incr(x1) weight: 0 hd(x1) weight: 0 #cons(x1,x2) weight: 0 s(x1) weight: 0 adx(x1) weight: 0 zeros() weight: 0 #tl(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 tl(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #incr(x1) weight: x1 #hd(x1) weight: 0 nats() weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { } Removed DPs: #8 #20 Number of SCCs: 6, DPs: 20, edges: 61 SCC { #7 #15 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #adx(x1) weight: 0 incr(x1) weight: 0 hd(x1) weight: 0 #cons(x1,x2) weight: 0 s(x1) weight: 0 adx(x1) weight: 0 zeros() weight: 0 #tl(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: x1 tl(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #incr(x1) weight: 0 #hd(x1) weight: 0 nats() weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { } Removed DPs: #7 #15 Number of SCCs: 5, DPs: 18, edges: 57 SCC { #3 #19 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #adx(x1) weight: 0 incr(x1) weight: 0 hd(x1) weight: 0 #cons(x1,x2) weight: 0 s(x1) weight: 0 adx(x1) weight: 0 zeros() weight: 0 #tl(x1) weight: x1 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 tl(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #incr(x1) weight: 0 #hd(x1) weight: 0 nats() weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { } Removed DPs: #3 #19 Number of SCCs: 4, DPs: 16, edges: 53 SCC { #22 #26 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #adx(x1) weight: x1 incr(x1) weight: 0 hd(x1) weight: 0 #cons(x1,x2) weight: 0 s(x1) weight: 0 adx(x1) weight: 0 zeros() weight: 0 #tl(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 tl(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #incr(x1) weight: 0 #hd(x1) weight: 0 nats() weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { } Removed DPs: #22 #26 Number of SCCs: 3, DPs: 14, edges: 49 SCC { #21 #24 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #adx(x1) weight: 0 incr(x1) weight: 0 hd(x1) weight: 0 #cons(x1,x2) weight: 0 s(x1) weight: 0 adx(x1) weight: 0 zeros() weight: 0 #tl(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 tl(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #incr(x1) weight: 0 #hd(x1) weight: x1 nats() weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { } Removed DPs: #21 #24 Number of SCCs: 2, DPs: 12, edges: 45 SCC { #16 #23 #25 #43 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #adx(x1) weight: 0 incr(x1) weight: 0 hd(x1) weight: 0 #cons(x1,x2) weight: x2 s(x1) weight: 0 adx(x1) weight: 0 zeros() weight: 0 #tl(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 tl(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #incr(x1) weight: 0 #hd(x1) weight: 0 nats() weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { } Removed DPs: #23 #25 Number of SCCs: 2, DPs: 10, edges: 33 SCC { #16 #43 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #adx(x1) weight: 0 incr(x1) weight: 0 hd(x1) weight: 0 #cons(x1,x2) weight: x1 s(x1) weight: 0 adx(x1) weight: 0 zeros() weight: 0 #tl(x1) weight: 0 #mark(x1) weight: 0 0() weight: 0 #s(x1) weight: 0 tl(x1) weight: 0 mark(x1) weight: (/ 1 2) + x1 #incr(x1) weight: 0 #hd(x1) weight: 0 nats() weight: 0 active(x1) weight: (/ 1 2) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { } Removed DPs: #16 #43 Number of SCCs: 1, DPs: 8, edges: 29 SCC { #11 #14 #17 #31 #33 #35 #38 #39 } Removing DPs: Order(PosReal,>,Sum)... succeeded. #adx(x1) weight: 0 incr(x1) weight: x1 hd(x1) weight: (/ 1 4) + x1 #cons(x1,x2) weight: 0 s(x1) weight: (/ 1 2) adx(x1) weight: x1 zeros() weight: 0 #tl(x1) weight: 0 #mark(x1) weight: x1 0() weight: 0 #s(x1) weight: 0 tl(x1) weight: (/ 1 4) + x1 mark(x1) weight: (/ 1 2) #incr(x1) weight: 0 #hd(x1) weight: 0 nats() weight: 0 active(x1) weight: (/ 1 4) + x1 cons(x1,x2) weight: 0 #active(x1) weight: 0 Usable rules: { 18..21 24 25 } Removed DPs: #14 #38 Number of SCCs: 2, DPs: 6, edges: 13 SCC { #11 #17 #31 #33 #35 #39 } Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded. #adx(x1) weight: 0 incr(x1) weight: (/ 1 4) + x1 hd(x1) weight: (/ 1 4) #cons(x1,x2) weight: 0 s(x1) weight: (/ 1 4) adx(x1) weight: (/ 1 2) + x1 zeros() weight: 0 #tl(x1) weight: 0 #mark(x1) weight: x1 0() weight: 0 #s(x1) weight: 0 tl(x1) weight: (/ 1 4) mark(x1) weight: (/ 1 4) #incr(x1) weight: 0 #hd(x1) weight: 0 nats() weight: (/ 3 4) active(x1) weight: (/ 1 4) cons(x1,x2) weight: 0 #active(x1) weight: (/ 1 2) Usable rules: { 1..4 7..15 18..21 28 29 } Removed DPs: #11 #17 #35 #39 Number of SCCs: 1, DPs: 0, edges: 0 YES