MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: activate(X) -> X activate(n__adx(X)) -> adx(X) activate(n__incr(X)) -> incr(X) activate(n__zeros()) -> zeros() adx(X) -> n__adx(X) adx(cons(X,L)) -> incr(cons(X,n__adx(activate(L)))) adx(nil()) -> nil() head(cons(X,L)) -> X incr(X) -> n__incr(X) incr(cons(X,L)) -> cons(s(X),n__incr(activate(L))) incr(nil()) -> nil() nats() -> adx(zeros()) tail(cons(X,L)) -> activate(L) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,adx/1,head/1,incr/1,nats/0,tail/1,zeros/0} / {0/0,cons/2,n__adx/1,n__incr/1,n__zeros/0,nil/0 ,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,adx,head,incr,nats,tail ,zeros} and constructors {0,cons,n__adx,n__incr,n__zeros,nil,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__adx(X)) -> c_2(adx#(X)) activate#(n__incr(X)) -> c_3(incr#(X)) activate#(n__zeros()) -> c_4(zeros#()) adx#(X) -> c_5() adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) adx#(nil()) -> c_7() head#(cons(X,L)) -> c_8() incr#(X) -> c_9() incr#(cons(X,L)) -> c_10(activate#(L)) incr#(nil()) -> c_11() nats#() -> c_12(adx#(zeros()),zeros#()) tail#(cons(X,L)) -> c_13(activate#(L)) zeros#() -> c_14() zeros#() -> c_15() Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__adx(X)) -> c_2(adx#(X)) activate#(n__incr(X)) -> c_3(incr#(X)) activate#(n__zeros()) -> c_4(zeros#()) adx#(X) -> c_5() adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) adx#(nil()) -> c_7() head#(cons(X,L)) -> c_8() incr#(X) -> c_9() incr#(cons(X,L)) -> c_10(activate#(L)) incr#(nil()) -> c_11() nats#() -> c_12(adx#(zeros()),zeros#()) tail#(cons(X,L)) -> c_13(activate#(L)) zeros#() -> c_14() zeros#() -> c_15() - Weak TRS: activate(X) -> X activate(n__adx(X)) -> adx(X) activate(n__incr(X)) -> incr(X) activate(n__zeros()) -> zeros() adx(X) -> n__adx(X) adx(cons(X,L)) -> incr(cons(X,n__adx(activate(L)))) adx(nil()) -> nil() head(cons(X,L)) -> X incr(X) -> n__incr(X) incr(cons(X,L)) -> cons(s(X),n__incr(activate(L))) incr(nil()) -> nil() nats() -> adx(zeros()) tail(cons(X,L)) -> activate(L) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,adx/1,head/1,incr/1,nats/0,tail/1,zeros/0,activate#/1,adx#/1,head#/1,incr#/1,nats#/0,tail#/1 ,zeros#/0} / {0/0,cons/2,n__adx/1,n__incr/1,n__zeros/0,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,adx#,head#,incr#,nats#,tail# ,zeros#} and constructors {0,cons,n__adx,n__incr,n__zeros,nil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__adx(X)) -> adx(X) activate(n__incr(X)) -> incr(X) activate(n__zeros()) -> zeros() adx(X) -> n__adx(X) adx(cons(X,L)) -> incr(cons(X,n__adx(activate(L)))) adx(nil()) -> nil() incr(X) -> n__incr(X) incr(cons(X,L)) -> cons(s(X),n__incr(activate(L))) incr(nil()) -> nil() zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() activate#(X) -> c_1() activate#(n__adx(X)) -> c_2(adx#(X)) activate#(n__incr(X)) -> c_3(incr#(X)) activate#(n__zeros()) -> c_4(zeros#()) adx#(X) -> c_5() adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) adx#(nil()) -> c_7() head#(cons(X,L)) -> c_8() incr#(X) -> c_9() incr#(cons(X,L)) -> c_10(activate#(L)) incr#(nil()) -> c_11() nats#() -> c_12(adx#(zeros()),zeros#()) tail#(cons(X,L)) -> c_13(activate#(L)) zeros#() -> c_14() zeros#() -> c_15() * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__adx(X)) -> c_2(adx#(X)) activate#(n__incr(X)) -> c_3(incr#(X)) activate#(n__zeros()) -> c_4(zeros#()) adx#(X) -> c_5() adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) adx#(nil()) -> c_7() head#(cons(X,L)) -> c_8() incr#(X) -> c_9() incr#(cons(X,L)) -> c_10(activate#(L)) incr#(nil()) -> c_11() nats#() -> c_12(adx#(zeros()),zeros#()) tail#(cons(X,L)) -> c_13(activate#(L)) zeros#() -> c_14() zeros#() -> c_15() - Weak TRS: activate(X) -> X activate(n__adx(X)) -> adx(X) activate(n__incr(X)) -> incr(X) activate(n__zeros()) -> zeros() adx(X) -> n__adx(X) adx(cons(X,L)) -> incr(cons(X,n__adx(activate(L)))) adx(nil()) -> nil() incr(X) -> n__incr(X) incr(cons(X,L)) -> cons(s(X),n__incr(activate(L))) incr(nil()) -> nil() zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,adx/1,head/1,incr/1,nats/0,tail/1,zeros/0,activate#/1,adx#/1,head#/1,incr#/1,nats#/0,tail#/1 ,zeros#/0} / {0/0,cons/2,n__adx/1,n__incr/1,n__zeros/0,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,adx#,head#,incr#,nats#,tail# ,zeros#} and constructors {0,cons,n__adx,n__incr,n__zeros,nil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,5,7,8,9,11,14,15} by application of Pre({1,5,7,8,9,11,14,15}) = {2,3,4,6,10,12,13}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__adx(X)) -> c_2(adx#(X)) 3: activate#(n__incr(X)) -> c_3(incr#(X)) 4: activate#(n__zeros()) -> c_4(zeros#()) 5: adx#(X) -> c_5() 6: adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) 7: adx#(nil()) -> c_7() 8: head#(cons(X,L)) -> c_8() 9: incr#(X) -> c_9() 10: incr#(cons(X,L)) -> c_10(activate#(L)) 11: incr#(nil()) -> c_11() 12: nats#() -> c_12(adx#(zeros()),zeros#()) 13: tail#(cons(X,L)) -> c_13(activate#(L)) 14: zeros#() -> c_14() 15: zeros#() -> c_15() * Step 4: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: activate#(n__adx(X)) -> c_2(adx#(X)) activate#(n__incr(X)) -> c_3(incr#(X)) activate#(n__zeros()) -> c_4(zeros#()) adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) incr#(cons(X,L)) -> c_10(activate#(L)) nats#() -> c_12(adx#(zeros()),zeros#()) tail#(cons(X,L)) -> c_13(activate#(L)) - Weak DPs: activate#(X) -> c_1() adx#(X) -> c_5() adx#(nil()) -> c_7() head#(cons(X,L)) -> c_8() incr#(X) -> c_9() incr#(nil()) -> c_11() zeros#() -> c_14() zeros#() -> c_15() - Weak TRS: activate(X) -> X activate(n__adx(X)) -> adx(X) activate(n__incr(X)) -> incr(X) activate(n__zeros()) -> zeros() adx(X) -> n__adx(X) adx(cons(X,L)) -> incr(cons(X,n__adx(activate(L)))) adx(nil()) -> nil() incr(X) -> n__incr(X) incr(cons(X,L)) -> cons(s(X),n__incr(activate(L))) incr(nil()) -> nil() zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,adx/1,head/1,incr/1,nats/0,tail/1,zeros/0,activate#/1,adx#/1,head#/1,incr#/1,nats#/0,tail#/1 ,zeros#/0} / {0/0,cons/2,n__adx/1,n__incr/1,n__zeros/0,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,adx#,head#,incr#,nats#,tail# ,zeros#} and constructors {0,cons,n__adx,n__incr,n__zeros,nil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3} by application of Pre({3}) = {4,5,7}. Here rules are labelled as follows: 1: activate#(n__adx(X)) -> c_2(adx#(X)) 2: activate#(n__incr(X)) -> c_3(incr#(X)) 3: activate#(n__zeros()) -> c_4(zeros#()) 4: adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) 5: incr#(cons(X,L)) -> c_10(activate#(L)) 6: nats#() -> c_12(adx#(zeros()),zeros#()) 7: tail#(cons(X,L)) -> c_13(activate#(L)) 8: activate#(X) -> c_1() 9: adx#(X) -> c_5() 10: adx#(nil()) -> c_7() 11: head#(cons(X,L)) -> c_8() 12: incr#(X) -> c_9() 13: incr#(nil()) -> c_11() 14: zeros#() -> c_14() 15: zeros#() -> c_15() * Step 5: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: activate#(n__adx(X)) -> c_2(adx#(X)) activate#(n__incr(X)) -> c_3(incr#(X)) adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) incr#(cons(X,L)) -> c_10(activate#(L)) nats#() -> c_12(adx#(zeros()),zeros#()) tail#(cons(X,L)) -> c_13(activate#(L)) - Weak DPs: activate#(X) -> c_1() activate#(n__zeros()) -> c_4(zeros#()) adx#(X) -> c_5() adx#(nil()) -> c_7() head#(cons(X,L)) -> c_8() incr#(X) -> c_9() incr#(nil()) -> c_11() zeros#() -> c_14() zeros#() -> c_15() - Weak TRS: activate(X) -> X activate(n__adx(X)) -> adx(X) activate(n__incr(X)) -> incr(X) activate(n__zeros()) -> zeros() adx(X) -> n__adx(X) adx(cons(X,L)) -> incr(cons(X,n__adx(activate(L)))) adx(nil()) -> nil() incr(X) -> n__incr(X) incr(cons(X,L)) -> cons(s(X),n__incr(activate(L))) incr(nil()) -> nil() zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,adx/1,head/1,incr/1,nats/0,tail/1,zeros/0,activate#/1,adx#/1,head#/1,incr#/1,nats#/0,tail#/1 ,zeros#/0} / {0/0,cons/2,n__adx/1,n__incr/1,n__zeros/0,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,adx#,head#,incr#,nats#,tail# ,zeros#} and constructors {0,cons,n__adx,n__incr,n__zeros,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__adx(X)) -> c_2(adx#(X)) -->_1 adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)):3 -->_1 adx#(nil()) -> c_7():10 -->_1 adx#(X) -> c_5():9 2:S:activate#(n__incr(X)) -> c_3(incr#(X)) -->_1 incr#(cons(X,L)) -> c_10(activate#(L)):4 -->_1 incr#(nil()) -> c_11():13 -->_1 incr#(X) -> c_9():12 3:S:adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) -->_2 activate#(n__zeros()) -> c_4(zeros#()):8 -->_1 incr#(cons(X,L)) -> c_10(activate#(L)):4 -->_1 incr#(X) -> c_9():12 -->_2 activate#(X) -> c_1():7 -->_2 activate#(n__incr(X)) -> c_3(incr#(X)):2 -->_2 activate#(n__adx(X)) -> c_2(adx#(X)):1 4:S:incr#(cons(X,L)) -> c_10(activate#(L)) -->_1 activate#(n__zeros()) -> c_4(zeros#()):8 -->_1 activate#(X) -> c_1():7 -->_1 activate#(n__incr(X)) -> c_3(incr#(X)):2 -->_1 activate#(n__adx(X)) -> c_2(adx#(X)):1 5:S:nats#() -> c_12(adx#(zeros()),zeros#()) -->_2 zeros#() -> c_15():15 -->_2 zeros#() -> c_14():14 -->_1 adx#(nil()) -> c_7():10 -->_1 adx#(X) -> c_5():9 -->_1 adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)):3 6:S:tail#(cons(X,L)) -> c_13(activate#(L)) -->_1 activate#(n__zeros()) -> c_4(zeros#()):8 -->_1 activate#(X) -> c_1():7 -->_1 activate#(n__incr(X)) -> c_3(incr#(X)):2 -->_1 activate#(n__adx(X)) -> c_2(adx#(X)):1 7:W:activate#(X) -> c_1() 8:W:activate#(n__zeros()) -> c_4(zeros#()) -->_1 zeros#() -> c_15():15 -->_1 zeros#() -> c_14():14 9:W:adx#(X) -> c_5() 10:W:adx#(nil()) -> c_7() 11:W:head#(cons(X,L)) -> c_8() 12:W:incr#(X) -> c_9() 13:W:incr#(nil()) -> c_11() 14:W:zeros#() -> c_14() 15:W:zeros#() -> c_15() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: head#(cons(X,L)) -> c_8() 9: adx#(X) -> c_5() 10: adx#(nil()) -> c_7() 12: incr#(X) -> c_9() 13: incr#(nil()) -> c_11() 7: activate#(X) -> c_1() 8: activate#(n__zeros()) -> c_4(zeros#()) 14: zeros#() -> c_14() 15: zeros#() -> c_15() * Step 6: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: activate#(n__adx(X)) -> c_2(adx#(X)) activate#(n__incr(X)) -> c_3(incr#(X)) adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) incr#(cons(X,L)) -> c_10(activate#(L)) nats#() -> c_12(adx#(zeros()),zeros#()) tail#(cons(X,L)) -> c_13(activate#(L)) - Weak TRS: activate(X) -> X activate(n__adx(X)) -> adx(X) activate(n__incr(X)) -> incr(X) activate(n__zeros()) -> zeros() adx(X) -> n__adx(X) adx(cons(X,L)) -> incr(cons(X,n__adx(activate(L)))) adx(nil()) -> nil() incr(X) -> n__incr(X) incr(cons(X,L)) -> cons(s(X),n__incr(activate(L))) incr(nil()) -> nil() zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,adx/1,head/1,incr/1,nats/0,tail/1,zeros/0,activate#/1,adx#/1,head#/1,incr#/1,nats#/0,tail#/1 ,zeros#/0} / {0/0,cons/2,n__adx/1,n__incr/1,n__zeros/0,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,adx#,head#,incr#,nats#,tail# ,zeros#} and constructors {0,cons,n__adx,n__incr,n__zeros,nil,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__adx(X)) -> c_2(adx#(X)) -->_1 adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)):3 2:S:activate#(n__incr(X)) -> c_3(incr#(X)) -->_1 incr#(cons(X,L)) -> c_10(activate#(L)):4 3:S:adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) -->_1 incr#(cons(X,L)) -> c_10(activate#(L)):4 -->_2 activate#(n__incr(X)) -> c_3(incr#(X)):2 -->_2 activate#(n__adx(X)) -> c_2(adx#(X)):1 4:S:incr#(cons(X,L)) -> c_10(activate#(L)) -->_1 activate#(n__incr(X)) -> c_3(incr#(X)):2 -->_1 activate#(n__adx(X)) -> c_2(adx#(X)):1 5:S:nats#() -> c_12(adx#(zeros()),zeros#()) -->_1 adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)):3 6:S:tail#(cons(X,L)) -> c_13(activate#(L)) -->_1 activate#(n__incr(X)) -> c_3(incr#(X)):2 -->_1 activate#(n__adx(X)) -> c_2(adx#(X)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: nats#() -> c_12(adx#(zeros())) * Step 7: RemoveHeads MAYBE + Considered Problem: - Strict DPs: activate#(n__adx(X)) -> c_2(adx#(X)) activate#(n__incr(X)) -> c_3(incr#(X)) adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) incr#(cons(X,L)) -> c_10(activate#(L)) nats#() -> c_12(adx#(zeros())) tail#(cons(X,L)) -> c_13(activate#(L)) - Weak TRS: activate(X) -> X activate(n__adx(X)) -> adx(X) activate(n__incr(X)) -> incr(X) activate(n__zeros()) -> zeros() adx(X) -> n__adx(X) adx(cons(X,L)) -> incr(cons(X,n__adx(activate(L)))) adx(nil()) -> nil() incr(X) -> n__incr(X) incr(cons(X,L)) -> cons(s(X),n__incr(activate(L))) incr(nil()) -> nil() zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,adx/1,head/1,incr/1,nats/0,tail/1,zeros/0,activate#/1,adx#/1,head#/1,incr#/1,nats#/0,tail#/1 ,zeros#/0} / {0/0,cons/2,n__adx/1,n__incr/1,n__zeros/0,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,adx#,head#,incr#,nats#,tail# ,zeros#} and constructors {0,cons,n__adx,n__incr,n__zeros,nil,s} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:activate#(n__adx(X)) -> c_2(adx#(X)) -->_1 adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)):3 2:S:activate#(n__incr(X)) -> c_3(incr#(X)) -->_1 incr#(cons(X,L)) -> c_10(activate#(L)):4 3:S:adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) -->_1 incr#(cons(X,L)) -> c_10(activate#(L)):4 -->_2 activate#(n__incr(X)) -> c_3(incr#(X)):2 -->_2 activate#(n__adx(X)) -> c_2(adx#(X)):1 4:S:incr#(cons(X,L)) -> c_10(activate#(L)) -->_1 activate#(n__incr(X)) -> c_3(incr#(X)):2 -->_1 activate#(n__adx(X)) -> c_2(adx#(X)):1 5:S:nats#() -> c_12(adx#(zeros())) -->_1 adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)):3 6:S:tail#(cons(X,L)) -> c_13(activate#(L)) -->_1 activate#(n__incr(X)) -> c_3(incr#(X)):2 -->_1 activate#(n__adx(X)) -> c_2(adx#(X)):1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(6,tail#(cons(X,L)) -> c_13(activate#(L)))] * Step 8: NaturalMI MAYBE + Considered Problem: - Strict DPs: activate#(n__adx(X)) -> c_2(adx#(X)) activate#(n__incr(X)) -> c_3(incr#(X)) adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) incr#(cons(X,L)) -> c_10(activate#(L)) nats#() -> c_12(adx#(zeros())) - Weak TRS: activate(X) -> X activate(n__adx(X)) -> adx(X) activate(n__incr(X)) -> incr(X) activate(n__zeros()) -> zeros() adx(X) -> n__adx(X) adx(cons(X,L)) -> incr(cons(X,n__adx(activate(L)))) adx(nil()) -> nil() incr(X) -> n__incr(X) incr(cons(X,L)) -> cons(s(X),n__incr(activate(L))) incr(nil()) -> nil() zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,adx/1,head/1,incr/1,nats/0,tail/1,zeros/0,activate#/1,adx#/1,head#/1,incr#/1,nats#/0,tail#/1 ,zeros#/0} / {0/0,cons/2,n__adx/1,n__incr/1,n__zeros/0,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,adx#,head#,incr#,nats#,tail# ,zeros#} and constructors {0,cons,n__adx,n__incr,n__zeros,nil,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {1,2}, uargs(c_10) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {activate#,adx#,head#,incr#,nats#,tail#,zeros#} TcT has computed the following interpretation: p(0) = [0] p(activate) = [10] p(adx) = [9] p(cons) = [0] p(head) = [0] p(incr) = [0] p(n__adx) = [0] p(n__incr) = [0] p(n__zeros) = [0] p(nats) = [0] p(nil) = [0] p(s) = [0] p(tail) = [0] p(zeros) = [0] p(activate#) = [0] p(adx#) = [0] p(head#) = [1] p(incr#) = [0] p(nats#) = [10] p(tail#) = [0] p(zeros#) = [1] p(c_1) = [1] p(c_2) = [8] x1 + [0] p(c_3) = [8] x1 + [0] p(c_4) = [2] p(c_5) = [1] p(c_6) = [1] x1 + [8] x2 + [0] p(c_7) = [1] p(c_8) = [4] p(c_9) = [0] p(c_10) = [2] x1 + [0] p(c_11) = [1] p(c_12) = [1] x1 + [1] p(c_13) = [1] x1 + [0] p(c_14) = [2] p(c_15) = [1] Following rules are strictly oriented: nats#() = [10] > [1] = c_12(adx#(zeros())) Following rules are (at-least) weakly oriented: activate#(n__adx(X)) = [0] >= [0] = c_2(adx#(X)) activate#(n__incr(X)) = [0] >= [0] = c_3(incr#(X)) adx#(cons(X,L)) = [0] >= [0] = c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) incr#(cons(X,L)) = [0] >= [0] = c_10(activate#(L)) * Step 9: Failure MAYBE + Considered Problem: - Strict DPs: activate#(n__adx(X)) -> c_2(adx#(X)) activate#(n__incr(X)) -> c_3(incr#(X)) adx#(cons(X,L)) -> c_6(incr#(cons(X,n__adx(activate(L)))),activate#(L)) incr#(cons(X,L)) -> c_10(activate#(L)) - Weak DPs: nats#() -> c_12(adx#(zeros())) - Weak TRS: activate(X) -> X activate(n__adx(X)) -> adx(X) activate(n__incr(X)) -> incr(X) activate(n__zeros()) -> zeros() adx(X) -> n__adx(X) adx(cons(X,L)) -> incr(cons(X,n__adx(activate(L)))) adx(nil()) -> nil() incr(X) -> n__incr(X) incr(cons(X,L)) -> cons(s(X),n__incr(activate(L))) incr(nil()) -> nil() zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,adx/1,head/1,incr/1,nats/0,tail/1,zeros/0,activate#/1,adx#/1,head#/1,incr#/1,nats#/0,tail#/1 ,zeros#/0} / {0/0,cons/2,n__adx/1,n__incr/1,n__zeros/0,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/2,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,adx#,head#,incr#,nats#,tail# ,zeros#} and constructors {0,cons,n__adx,n__incr,n__zeros,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE