MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: empty(cons(n,x)) -> false() empty(nil()) -> true() head(cons(n,x)) -> n if(false(),b,x) -> if2(b,x) if(true(),b,x) -> weight_undefined_error() if2(false(),x) -> weight(sum(x,cons(0(),tail(tail(x))))) if2(true(),x) -> head(x) sum(cons(0(),x),y) -> sum(x,y) sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y)) sum(nil(),y) -> y tail(cons(n,x)) -> x tail(nil()) -> nil() weight(x) -> if(empty(x),empty(tail(x)),x) - Signature: {empty/1,head/1,if/3,if2/2,sum/2,tail/1,weight/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0 ,weight_undefined_error/0} - Obligation: innermost runtime complexity wrt. defined symbols {empty,head,if,if2,sum,tail,weight} and constructors {0 ,cons,false,nil,s,true,weight_undefined_error} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs empty#(cons(n,x)) -> c_1() empty#(nil()) -> c_2() head#(cons(n,x)) -> c_3() if#(false(),b,x) -> c_4(if2#(b,x)) if#(true(),b,x) -> c_5() if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))) ,sum#(x,cons(0(),tail(tail(x)))) ,tail#(tail(x)) ,tail#(x)) if2#(true(),x) -> c_7(head#(x)) sum#(cons(0(),x),y) -> c_8(sum#(x,y)) sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))) sum#(nil(),y) -> c_10() tail#(cons(n,x)) -> c_11() tail#(nil()) -> c_12() weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x),empty#(x),empty#(tail(x)),tail#(x)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: empty#(cons(n,x)) -> c_1() empty#(nil()) -> c_2() head#(cons(n,x)) -> c_3() if#(false(),b,x) -> c_4(if2#(b,x)) if#(true(),b,x) -> c_5() if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))) ,sum#(x,cons(0(),tail(tail(x)))) ,tail#(tail(x)) ,tail#(x)) if2#(true(),x) -> c_7(head#(x)) sum#(cons(0(),x),y) -> c_8(sum#(x,y)) sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))) sum#(nil(),y) -> c_10() tail#(cons(n,x)) -> c_11() tail#(nil()) -> c_12() weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x),empty#(x),empty#(tail(x)),tail#(x)) - Weak TRS: empty(cons(n,x)) -> false() empty(nil()) -> true() head(cons(n,x)) -> n if(false(),b,x) -> if2(b,x) if(true(),b,x) -> weight_undefined_error() if2(false(),x) -> weight(sum(x,cons(0(),tail(tail(x))))) if2(true(),x) -> head(x) sum(cons(0(),x),y) -> sum(x,y) sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y)) sum(nil(),y) -> y tail(cons(n,x)) -> x tail(nil()) -> nil() weight(x) -> if(empty(x),empty(tail(x)),x) - Signature: {empty/1,head/1,if/3,if2/2,sum/2,tail/1,weight/1,empty#/1,head#/1,if#/3,if2#/2,sum#/2,tail#/1 ,weight#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,weight_undefined_error/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0 ,c_6/4,c_7/1,c_8/1,c_9/1,c_10/0,c_11/0,c_12/0,c_13/4} - Obligation: innermost runtime complexity wrt. defined symbols {empty#,head#,if#,if2#,sum#,tail# ,weight#} and constructors {0,cons,false,nil,s,true,weight_undefined_error} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: empty(cons(n,x)) -> false() empty(nil()) -> true() sum(cons(0(),x),y) -> sum(x,y) sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y)) sum(nil(),y) -> y tail(cons(n,x)) -> x tail(nil()) -> nil() empty#(cons(n,x)) -> c_1() empty#(nil()) -> c_2() head#(cons(n,x)) -> c_3() if#(false(),b,x) -> c_4(if2#(b,x)) if#(true(),b,x) -> c_5() if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))) ,sum#(x,cons(0(),tail(tail(x)))) ,tail#(tail(x)) ,tail#(x)) if2#(true(),x) -> c_7(head#(x)) sum#(cons(0(),x),y) -> c_8(sum#(x,y)) sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))) sum#(nil(),y) -> c_10() tail#(cons(n,x)) -> c_11() tail#(nil()) -> c_12() weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x),empty#(x),empty#(tail(x)),tail#(x)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: empty#(cons(n,x)) -> c_1() empty#(nil()) -> c_2() head#(cons(n,x)) -> c_3() if#(false(),b,x) -> c_4(if2#(b,x)) if#(true(),b,x) -> c_5() if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))) ,sum#(x,cons(0(),tail(tail(x)))) ,tail#(tail(x)) ,tail#(x)) if2#(true(),x) -> c_7(head#(x)) sum#(cons(0(),x),y) -> c_8(sum#(x,y)) sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))) sum#(nil(),y) -> c_10() tail#(cons(n,x)) -> c_11() tail#(nil()) -> c_12() weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x),empty#(x),empty#(tail(x)),tail#(x)) - Weak TRS: empty(cons(n,x)) -> false() empty(nil()) -> true() sum(cons(0(),x),y) -> sum(x,y) sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y)) sum(nil(),y) -> y tail(cons(n,x)) -> x tail(nil()) -> nil() - Signature: {empty/1,head/1,if/3,if2/2,sum/2,tail/1,weight/1,empty#/1,head#/1,if#/3,if2#/2,sum#/2,tail#/1 ,weight#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,weight_undefined_error/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0 ,c_6/4,c_7/1,c_8/1,c_9/1,c_10/0,c_11/0,c_12/0,c_13/4} - Obligation: innermost runtime complexity wrt. defined symbols {empty#,head#,if#,if2#,sum#,tail# ,weight#} and constructors {0,cons,false,nil,s,true,weight_undefined_error} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,5,10,11,12} by application of Pre({1,2,3,5,10,11,12}) = {6,7,8,13}. Here rules are labelled as follows: 1: empty#(cons(n,x)) -> c_1() 2: empty#(nil()) -> c_2() 3: head#(cons(n,x)) -> c_3() 4: if#(false(),b,x) -> c_4(if2#(b,x)) 5: if#(true(),b,x) -> c_5() 6: if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))) ,sum#(x,cons(0(),tail(tail(x)))) ,tail#(tail(x)) ,tail#(x)) 7: if2#(true(),x) -> c_7(head#(x)) 8: sum#(cons(0(),x),y) -> c_8(sum#(x,y)) 9: sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))) 10: sum#(nil(),y) -> c_10() 11: tail#(cons(n,x)) -> c_11() 12: tail#(nil()) -> c_12() 13: weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x),empty#(x),empty#(tail(x)),tail#(x)) * Step 4: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: if#(false(),b,x) -> c_4(if2#(b,x)) if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))) ,sum#(x,cons(0(),tail(tail(x)))) ,tail#(tail(x)) ,tail#(x)) if2#(true(),x) -> c_7(head#(x)) sum#(cons(0(),x),y) -> c_8(sum#(x,y)) sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))) weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x),empty#(x),empty#(tail(x)),tail#(x)) - Weak DPs: empty#(cons(n,x)) -> c_1() empty#(nil()) -> c_2() head#(cons(n,x)) -> c_3() if#(true(),b,x) -> c_5() sum#(nil(),y) -> c_10() tail#(cons(n,x)) -> c_11() tail#(nil()) -> c_12() - Weak TRS: empty(cons(n,x)) -> false() empty(nil()) -> true() sum(cons(0(),x),y) -> sum(x,y) sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y)) sum(nil(),y) -> y tail(cons(n,x)) -> x tail(nil()) -> nil() - Signature: {empty/1,head/1,if/3,if2/2,sum/2,tail/1,weight/1,empty#/1,head#/1,if#/3,if2#/2,sum#/2,tail#/1 ,weight#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,weight_undefined_error/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0 ,c_6/4,c_7/1,c_8/1,c_9/1,c_10/0,c_11/0,c_12/0,c_13/4} - Obligation: innermost runtime complexity wrt. defined symbols {empty#,head#,if#,if2#,sum#,tail# ,weight#} and constructors {0,cons,false,nil,s,true,weight_undefined_error} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3} by application of Pre({3}) = {1}. Here rules are labelled as follows: 1: if#(false(),b,x) -> c_4(if2#(b,x)) 2: if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))) ,sum#(x,cons(0(),tail(tail(x)))) ,tail#(tail(x)) ,tail#(x)) 3: if2#(true(),x) -> c_7(head#(x)) 4: sum#(cons(0(),x),y) -> c_8(sum#(x,y)) 5: sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))) 6: weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x),empty#(x),empty#(tail(x)),tail#(x)) 7: empty#(cons(n,x)) -> c_1() 8: empty#(nil()) -> c_2() 9: head#(cons(n,x)) -> c_3() 10: if#(true(),b,x) -> c_5() 11: sum#(nil(),y) -> c_10() 12: tail#(cons(n,x)) -> c_11() 13: tail#(nil()) -> c_12() * Step 5: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: if#(false(),b,x) -> c_4(if2#(b,x)) if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))) ,sum#(x,cons(0(),tail(tail(x)))) ,tail#(tail(x)) ,tail#(x)) sum#(cons(0(),x),y) -> c_8(sum#(x,y)) sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))) weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x),empty#(x),empty#(tail(x)),tail#(x)) - Weak DPs: empty#(cons(n,x)) -> c_1() empty#(nil()) -> c_2() head#(cons(n,x)) -> c_3() if#(true(),b,x) -> c_5() if2#(true(),x) -> c_7(head#(x)) sum#(nil(),y) -> c_10() tail#(cons(n,x)) -> c_11() tail#(nil()) -> c_12() - Weak TRS: empty(cons(n,x)) -> false() empty(nil()) -> true() sum(cons(0(),x),y) -> sum(x,y) sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y)) sum(nil(),y) -> y tail(cons(n,x)) -> x tail(nil()) -> nil() - Signature: {empty/1,head/1,if/3,if2/2,sum/2,tail/1,weight/1,empty#/1,head#/1,if#/3,if2#/2,sum#/2,tail#/1 ,weight#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,weight_undefined_error/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0 ,c_6/4,c_7/1,c_8/1,c_9/1,c_10/0,c_11/0,c_12/0,c_13/4} - Obligation: innermost runtime complexity wrt. defined symbols {empty#,head#,if#,if2#,sum#,tail# ,weight#} and constructors {0,cons,false,nil,s,true,weight_undefined_error} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:if#(false(),b,x) -> c_4(if2#(b,x)) -->_1 if2#(true(),x) -> c_7(head#(x)):10 -->_1 if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))) ,sum#(x,cons(0(),tail(tail(x)))) ,tail#(tail(x)) ,tail#(x)):2 2:S:if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))) ,sum#(x,cons(0(),tail(tail(x)))) ,tail#(tail(x)) ,tail#(x)) -->_1 weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x),empty#(x),empty#(tail(x)),tail#(x)):5 -->_2 sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))):4 -->_2 sum#(cons(0(),x),y) -> c_8(sum#(x,y)):3 -->_4 tail#(nil()) -> c_12():13 -->_3 tail#(nil()) -> c_12():13 -->_4 tail#(cons(n,x)) -> c_11():12 -->_3 tail#(cons(n,x)) -> c_11():12 -->_2 sum#(nil(),y) -> c_10():11 3:S:sum#(cons(0(),x),y) -> c_8(sum#(x,y)) -->_1 sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))):4 -->_1 sum#(nil(),y) -> c_10():11 -->_1 sum#(cons(0(),x),y) -> c_8(sum#(x,y)):3 4:S:sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))) -->_1 sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))):4 -->_1 sum#(cons(0(),x),y) -> c_8(sum#(x,y)):3 5:S:weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x),empty#(x),empty#(tail(x)),tail#(x)) -->_4 tail#(nil()) -> c_12():13 -->_4 tail#(cons(n,x)) -> c_11():12 -->_1 if#(true(),b,x) -> c_5():9 -->_3 empty#(nil()) -> c_2():7 -->_2 empty#(nil()) -> c_2():7 -->_3 empty#(cons(n,x)) -> c_1():6 -->_2 empty#(cons(n,x)) -> c_1():6 -->_1 if#(false(),b,x) -> c_4(if2#(b,x)):1 6:W:empty#(cons(n,x)) -> c_1() 7:W:empty#(nil()) -> c_2() 8:W:head#(cons(n,x)) -> c_3() 9:W:if#(true(),b,x) -> c_5() 10:W:if2#(true(),x) -> c_7(head#(x)) -->_1 head#(cons(n,x)) -> c_3():8 11:W:sum#(nil(),y) -> c_10() 12:W:tail#(cons(n,x)) -> c_11() 13:W:tail#(nil()) -> c_12() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: sum#(nil(),y) -> c_10() 6: empty#(cons(n,x)) -> c_1() 7: empty#(nil()) -> c_2() 9: if#(true(),b,x) -> c_5() 12: tail#(cons(n,x)) -> c_11() 13: tail#(nil()) -> c_12() 10: if2#(true(),x) -> c_7(head#(x)) 8: head#(cons(n,x)) -> c_3() * Step 6: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: if#(false(),b,x) -> c_4(if2#(b,x)) if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))) ,sum#(x,cons(0(),tail(tail(x)))) ,tail#(tail(x)) ,tail#(x)) sum#(cons(0(),x),y) -> c_8(sum#(x,y)) sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))) weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x),empty#(x),empty#(tail(x)),tail#(x)) - Weak TRS: empty(cons(n,x)) -> false() empty(nil()) -> true() sum(cons(0(),x),y) -> sum(x,y) sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y)) sum(nil(),y) -> y tail(cons(n,x)) -> x tail(nil()) -> nil() - Signature: {empty/1,head/1,if/3,if2/2,sum/2,tail/1,weight/1,empty#/1,head#/1,if#/3,if2#/2,sum#/2,tail#/1 ,weight#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,weight_undefined_error/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0 ,c_6/4,c_7/1,c_8/1,c_9/1,c_10/0,c_11/0,c_12/0,c_13/4} - Obligation: innermost runtime complexity wrt. defined symbols {empty#,head#,if#,if2#,sum#,tail# ,weight#} and constructors {0,cons,false,nil,s,true,weight_undefined_error} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:if#(false(),b,x) -> c_4(if2#(b,x)) -->_1 if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))) ,sum#(x,cons(0(),tail(tail(x)))) ,tail#(tail(x)) ,tail#(x)):2 2:S:if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))) ,sum#(x,cons(0(),tail(tail(x)))) ,tail#(tail(x)) ,tail#(x)) -->_1 weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x),empty#(x),empty#(tail(x)),tail#(x)):5 -->_2 sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))):4 -->_2 sum#(cons(0(),x),y) -> c_8(sum#(x,y)):3 3:S:sum#(cons(0(),x),y) -> c_8(sum#(x,y)) -->_1 sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))):4 -->_1 sum#(cons(0(),x),y) -> c_8(sum#(x,y)):3 4:S:sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))) -->_1 sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))):4 -->_1 sum#(cons(0(),x),y) -> c_8(sum#(x,y)):3 5:S:weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x),empty#(x),empty#(tail(x)),tail#(x)) -->_1 if#(false(),b,x) -> c_4(if2#(b,x)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))),sum#(x,cons(0(),tail(tail(x))))) weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x)) * Step 7: Failure MAYBE + Considered Problem: - Strict DPs: if#(false(),b,x) -> c_4(if2#(b,x)) if2#(false(),x) -> c_6(weight#(sum(x,cons(0(),tail(tail(x))))),sum#(x,cons(0(),tail(tail(x))))) sum#(cons(0(),x),y) -> c_8(sum#(x,y)) sum#(cons(s(n),x),cons(m,y)) -> c_9(sum#(cons(n,x),cons(s(m),y))) weight#(x) -> c_13(if#(empty(x),empty(tail(x)),x)) - Weak TRS: empty(cons(n,x)) -> false() empty(nil()) -> true() sum(cons(0(),x),y) -> sum(x,y) sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y)) sum(nil(),y) -> y tail(cons(n,x)) -> x tail(nil()) -> nil() - Signature: {empty/1,head/1,if/3,if2/2,sum/2,tail/1,weight/1,empty#/1,head#/1,if#/3,if2#/2,sum#/2,tail#/1 ,weight#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,weight_undefined_error/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/0 ,c_6/2,c_7/1,c_8/1,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {empty#,head#,if#,if2#,sum#,tail# ,weight#} and constructors {0,cons,false,nil,s,true,weight_undefined_error} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE