MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: head(cons(x,xs)) -> x if(false(),false(),y,xs,ys,x) -> sumList(ys,x) if(false(),true(),y,xs,ys,x) -> sumList(xs,y) if(true(),b,y,xs,ys,x) -> y inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) isEmpty(cons(x,xs)) -> false() isEmpty(nil()) -> true() isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) sum(xs) -> sumList(xs,0()) sumList(xs,y) -> if(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) tail(cons(x,xs)) -> xs tail(nil()) -> nil() - Signature: {head/1,if/6,inc/1,isEmpty/1,isZero/1,p/1,sum/1,sumList/2,tail/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {head,if,inc,isEmpty,isZero,p,sum,sumList ,tail} and constructors {0,cons,false,nil,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs head#(cons(x,xs)) -> c_1() if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)) if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)) if#(true(),b,y,xs,ys,x) -> c_4() inc#(0()) -> c_5() inc#(s(x)) -> c_6(inc#(x)) isEmpty#(cons(x,xs)) -> c_7() isEmpty#(nil()) -> c_8() isZero#(0()) -> c_9() isZero#(s(x)) -> c_10() p#(0()) -> c_11() p#(s(0())) -> c_12() p#(s(s(x))) -> c_13(p#(s(x))) sum#(xs) -> c_14(sumList#(xs,0())) sumList#(xs,y) -> c_15(if#(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)) tail#(cons(x,xs)) -> c_16() tail#(nil()) -> c_17() Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: head#(cons(x,xs)) -> c_1() if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)) if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)) if#(true(),b,y,xs,ys,x) -> c_4() inc#(0()) -> c_5() inc#(s(x)) -> c_6(inc#(x)) isEmpty#(cons(x,xs)) -> c_7() isEmpty#(nil()) -> c_8() isZero#(0()) -> c_9() isZero#(s(x)) -> c_10() p#(0()) -> c_11() p#(s(0())) -> c_12() p#(s(s(x))) -> c_13(p#(s(x))) sum#(xs) -> c_14(sumList#(xs,0())) sumList#(xs,y) -> c_15(if#(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)) tail#(cons(x,xs)) -> c_16() tail#(nil()) -> c_17() - Weak TRS: head(cons(x,xs)) -> x if(false(),false(),y,xs,ys,x) -> sumList(ys,x) if(false(),true(),y,xs,ys,x) -> sumList(xs,y) if(true(),b,y,xs,ys,x) -> y inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) isEmpty(cons(x,xs)) -> false() isEmpty(nil()) -> true() isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) sum(xs) -> sumList(xs,0()) sumList(xs,y) -> if(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) tail(cons(x,xs)) -> xs tail(nil()) -> nil() - Signature: {head/1,if/6,inc/1,isEmpty/1,isZero/1,p/1,sum/1,sumList/2,tail/1,head#/1,if#/6,inc#/1,isEmpty#/1,isZero#/1 ,p#/1,sum#/1,sumList#/2,tail#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/9,c_16/0,c_17/0} - Obligation: innermost runtime complexity wrt. defined symbols {head#,if#,inc#,isEmpty#,isZero#,p#,sum#,sumList# ,tail#} and constructors {0,cons,false,nil,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: head(cons(x,xs)) -> x inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) isEmpty(cons(x,xs)) -> false() isEmpty(nil()) -> true() isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) tail(cons(x,xs)) -> xs tail(nil()) -> nil() head#(cons(x,xs)) -> c_1() if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)) if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)) if#(true(),b,y,xs,ys,x) -> c_4() inc#(0()) -> c_5() inc#(s(x)) -> c_6(inc#(x)) isEmpty#(cons(x,xs)) -> c_7() isEmpty#(nil()) -> c_8() isZero#(0()) -> c_9() isZero#(s(x)) -> c_10() p#(0()) -> c_11() p#(s(0())) -> c_12() p#(s(s(x))) -> c_13(p#(s(x))) sum#(xs) -> c_14(sumList#(xs,0())) sumList#(xs,y) -> c_15(if#(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)) tail#(cons(x,xs)) -> c_16() tail#(nil()) -> c_17() * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: head#(cons(x,xs)) -> c_1() if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)) if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)) if#(true(),b,y,xs,ys,x) -> c_4() inc#(0()) -> c_5() inc#(s(x)) -> c_6(inc#(x)) isEmpty#(cons(x,xs)) -> c_7() isEmpty#(nil()) -> c_8() isZero#(0()) -> c_9() isZero#(s(x)) -> c_10() p#(0()) -> c_11() p#(s(0())) -> c_12() p#(s(s(x))) -> c_13(p#(s(x))) sum#(xs) -> c_14(sumList#(xs,0())) sumList#(xs,y) -> c_15(if#(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)) tail#(cons(x,xs)) -> c_16() tail#(nil()) -> c_17() - Weak TRS: head(cons(x,xs)) -> x inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) isEmpty(cons(x,xs)) -> false() isEmpty(nil()) -> true() isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) tail(cons(x,xs)) -> xs tail(nil()) -> nil() - Signature: {head/1,if/6,inc/1,isEmpty/1,isZero/1,p/1,sum/1,sumList/2,tail/1,head#/1,if#/6,inc#/1,isEmpty#/1,isZero#/1 ,p#/1,sum#/1,sumList#/2,tail#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/9,c_16/0,c_17/0} - Obligation: innermost runtime complexity wrt. defined symbols {head#,if#,inc#,isEmpty#,isZero#,p#,sum#,sumList# ,tail#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,5,7,8,9,10,11,12,16,17} by application of Pre({1,4,5,7,8,9,10,11,12,16,17}) = {6,13,15}. Here rules are labelled as follows: 1: head#(cons(x,xs)) -> c_1() 2: if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)) 3: if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)) 4: if#(true(),b,y,xs,ys,x) -> c_4() 5: inc#(0()) -> c_5() 6: inc#(s(x)) -> c_6(inc#(x)) 7: isEmpty#(cons(x,xs)) -> c_7() 8: isEmpty#(nil()) -> c_8() 9: isZero#(0()) -> c_9() 10: isZero#(s(x)) -> c_10() 11: p#(0()) -> c_11() 12: p#(s(0())) -> c_12() 13: p#(s(s(x))) -> c_13(p#(s(x))) 14: sum#(xs) -> c_14(sumList#(xs,0())) 15: sumList#(xs,y) -> c_15(if#(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)) 16: tail#(cons(x,xs)) -> c_16() 17: tail#(nil()) -> c_17() * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)) if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)) inc#(s(x)) -> c_6(inc#(x)) p#(s(s(x))) -> c_13(p#(s(x))) sum#(xs) -> c_14(sumList#(xs,0())) sumList#(xs,y) -> c_15(if#(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)) - Weak DPs: head#(cons(x,xs)) -> c_1() if#(true(),b,y,xs,ys,x) -> c_4() inc#(0()) -> c_5() isEmpty#(cons(x,xs)) -> c_7() isEmpty#(nil()) -> c_8() isZero#(0()) -> c_9() isZero#(s(x)) -> c_10() p#(0()) -> c_11() p#(s(0())) -> c_12() tail#(cons(x,xs)) -> c_16() tail#(nil()) -> c_17() - Weak TRS: head(cons(x,xs)) -> x inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) isEmpty(cons(x,xs)) -> false() isEmpty(nil()) -> true() isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) tail(cons(x,xs)) -> xs tail(nil()) -> nil() - Signature: {head/1,if/6,inc/1,isEmpty/1,isZero/1,p/1,sum/1,sumList/2,tail/1,head#/1,if#/6,inc#/1,isEmpty#/1,isZero#/1 ,p#/1,sum#/1,sumList#/2,tail#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/9,c_16/0,c_17/0} - Obligation: innermost runtime complexity wrt. defined symbols {head#,if#,inc#,isEmpty#,isZero#,p#,sum#,sumList# ,tail#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)) -->_1 sumList#(xs,y) -> c_15(if#(isEmpty(xs) ,isZero(head(xs)) ,y ,tail(xs) ,cons(p(head(xs)),tail(xs)) ,inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)):6 2:S:if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)) -->_1 sumList#(xs,y) -> c_15(if#(isEmpty(xs) ,isZero(head(xs)) ,y ,tail(xs) ,cons(p(head(xs)),tail(xs)) ,inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)):6 3:S:inc#(s(x)) -> c_6(inc#(x)) -->_1 inc#(0()) -> c_5():9 -->_1 inc#(s(x)) -> c_6(inc#(x)):3 4:S:p#(s(s(x))) -> c_13(p#(s(x))) -->_1 p#(s(0())) -> c_12():15 -->_1 p#(s(s(x))) -> c_13(p#(s(x))):4 5:S:sum#(xs) -> c_14(sumList#(xs,0())) -->_1 sumList#(xs,y) -> c_15(if#(isEmpty(xs) ,isZero(head(xs)) ,y ,tail(xs) ,cons(p(head(xs)),tail(xs)) ,inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)):6 6:S:sumList#(xs,y) -> c_15(if#(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)) -->_8 tail#(nil()) -> c_17():17 -->_5 tail#(nil()) -> c_17():17 -->_8 tail#(cons(x,xs)) -> c_16():16 -->_5 tail#(cons(x,xs)) -> c_16():16 -->_6 p#(s(0())) -> c_12():15 -->_6 p#(0()) -> c_11():14 -->_3 isZero#(s(x)) -> c_10():13 -->_3 isZero#(0()) -> c_9():12 -->_2 isEmpty#(nil()) -> c_8():11 -->_2 isEmpty#(cons(x,xs)) -> c_7():10 -->_9 inc#(0()) -> c_5():9 -->_1 if#(true(),b,y,xs,ys,x) -> c_4():8 -->_7 head#(cons(x,xs)) -> c_1():7 -->_4 head#(cons(x,xs)) -> c_1():7 -->_6 p#(s(s(x))) -> c_13(p#(s(x))):4 -->_9 inc#(s(x)) -> c_6(inc#(x)):3 -->_1 if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)):2 -->_1 if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)):1 7:W:head#(cons(x,xs)) -> c_1() 8:W:if#(true(),b,y,xs,ys,x) -> c_4() 9:W:inc#(0()) -> c_5() 10:W:isEmpty#(cons(x,xs)) -> c_7() 11:W:isEmpty#(nil()) -> c_8() 12:W:isZero#(0()) -> c_9() 13:W:isZero#(s(x)) -> c_10() 14:W:p#(0()) -> c_11() 15:W:p#(s(0())) -> c_12() 16:W:tail#(cons(x,xs)) -> c_16() 17:W:tail#(nil()) -> c_17() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: head#(cons(x,xs)) -> c_1() 8: if#(true(),b,y,xs,ys,x) -> c_4() 9: inc#(0()) -> c_5() 10: isEmpty#(cons(x,xs)) -> c_7() 11: isEmpty#(nil()) -> c_8() 12: isZero#(0()) -> c_9() 13: isZero#(s(x)) -> c_10() 14: p#(0()) -> c_11() 15: p#(s(0())) -> c_12() 16: tail#(cons(x,xs)) -> c_16() 17: tail#(nil()) -> c_17() * Step 5: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)) if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)) inc#(s(x)) -> c_6(inc#(x)) p#(s(s(x))) -> c_13(p#(s(x))) sum#(xs) -> c_14(sumList#(xs,0())) sumList#(xs,y) -> c_15(if#(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)) - Weak TRS: head(cons(x,xs)) -> x inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) isEmpty(cons(x,xs)) -> false() isEmpty(nil()) -> true() isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) tail(cons(x,xs)) -> xs tail(nil()) -> nil() - Signature: {head/1,if/6,inc/1,isEmpty/1,isZero/1,p/1,sum/1,sumList/2,tail/1,head#/1,if#/6,inc#/1,isEmpty#/1,isZero#/1 ,p#/1,sum#/1,sumList#/2,tail#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/9,c_16/0,c_17/0} - Obligation: innermost runtime complexity wrt. defined symbols {head#,if#,inc#,isEmpty#,isZero#,p#,sum#,sumList# ,tail#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)) -->_1 sumList#(xs,y) -> c_15(if#(isEmpty(xs) ,isZero(head(xs)) ,y ,tail(xs) ,cons(p(head(xs)),tail(xs)) ,inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)):6 2:S:if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)) -->_1 sumList#(xs,y) -> c_15(if#(isEmpty(xs) ,isZero(head(xs)) ,y ,tail(xs) ,cons(p(head(xs)),tail(xs)) ,inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)):6 3:S:inc#(s(x)) -> c_6(inc#(x)) -->_1 inc#(s(x)) -> c_6(inc#(x)):3 4:S:p#(s(s(x))) -> c_13(p#(s(x))) -->_1 p#(s(s(x))) -> c_13(p#(s(x))):4 5:S:sum#(xs) -> c_14(sumList#(xs,0())) -->_1 sumList#(xs,y) -> c_15(if#(isEmpty(xs) ,isZero(head(xs)) ,y ,tail(xs) ,cons(p(head(xs)),tail(xs)) ,inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)):6 6:S:sumList#(xs,y) -> c_15(if#(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) ,isEmpty#(xs) ,isZero#(head(xs)) ,head#(xs) ,tail#(xs) ,p#(head(xs)) ,head#(xs) ,tail#(xs) ,inc#(y)) -->_6 p#(s(s(x))) -> c_13(p#(s(x))):4 -->_9 inc#(s(x)) -> c_6(inc#(x)):3 -->_1 if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)):2 -->_1 if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sumList#(xs,y) -> c_15(if#(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) ,p#(head(xs)) ,inc#(y)) * Step 6: RemoveHeads MAYBE + Considered Problem: - Strict DPs: if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)) if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)) inc#(s(x)) -> c_6(inc#(x)) p#(s(s(x))) -> c_13(p#(s(x))) sum#(xs) -> c_14(sumList#(xs,0())) sumList#(xs,y) -> c_15(if#(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) ,p#(head(xs)) ,inc#(y)) - Weak TRS: head(cons(x,xs)) -> x inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) isEmpty(cons(x,xs)) -> false() isEmpty(nil()) -> true() isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) tail(cons(x,xs)) -> xs tail(nil()) -> nil() - Signature: {head/1,if/6,inc/1,isEmpty/1,isZero/1,p/1,sum/1,sumList/2,tail/1,head#/1,if#/6,inc#/1,isEmpty#/1,isZero#/1 ,p#/1,sum#/1,sumList#/2,tail#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/3,c_16/0,c_17/0} - Obligation: innermost runtime complexity wrt. defined symbols {head#,if#,inc#,isEmpty#,isZero#,p#,sum#,sumList# ,tail#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)) -->_1 sumList#(xs,y) -> c_15(if#(isEmpty(xs) ,isZero(head(xs)) ,y ,tail(xs) ,cons(p(head(xs)),tail(xs)) ,inc(y)) ,p#(head(xs)) ,inc#(y)):6 2:S:if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)) -->_1 sumList#(xs,y) -> c_15(if#(isEmpty(xs) ,isZero(head(xs)) ,y ,tail(xs) ,cons(p(head(xs)),tail(xs)) ,inc(y)) ,p#(head(xs)) ,inc#(y)):6 3:S:inc#(s(x)) -> c_6(inc#(x)) -->_1 inc#(s(x)) -> c_6(inc#(x)):3 4:S:p#(s(s(x))) -> c_13(p#(s(x))) -->_1 p#(s(s(x))) -> c_13(p#(s(x))):4 5:S:sum#(xs) -> c_14(sumList#(xs,0())) -->_1 sumList#(xs,y) -> c_15(if#(isEmpty(xs) ,isZero(head(xs)) ,y ,tail(xs) ,cons(p(head(xs)),tail(xs)) ,inc(y)) ,p#(head(xs)) ,inc#(y)):6 6:S:sumList#(xs,y) -> c_15(if#(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) ,p#(head(xs)) ,inc#(y)) -->_2 p#(s(s(x))) -> c_13(p#(s(x))):4 -->_3 inc#(s(x)) -> c_6(inc#(x)):3 -->_1 if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)):2 -->_1 if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,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). [(5,sum#(xs) -> c_14(sumList#(xs,0())))] * Step 7: Failure MAYBE + Considered Problem: - Strict DPs: if#(false(),false(),y,xs,ys,x) -> c_2(sumList#(ys,x)) if#(false(),true(),y,xs,ys,x) -> c_3(sumList#(xs,y)) inc#(s(x)) -> c_6(inc#(x)) p#(s(s(x))) -> c_13(p#(s(x))) sumList#(xs,y) -> c_15(if#(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) ,p#(head(xs)) ,inc#(y)) - Weak TRS: head(cons(x,xs)) -> x inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) isEmpty(cons(x,xs)) -> false() isEmpty(nil()) -> true() isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) tail(cons(x,xs)) -> xs tail(nil()) -> nil() - Signature: {head/1,if/6,inc/1,isEmpty/1,isZero/1,p/1,sum/1,sumList/2,tail/1,head#/1,if#/6,inc#/1,isEmpty#/1,isZero#/1 ,p#/1,sum#/1,sumList#/2,tail#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/3,c_16/0,c_17/0} - Obligation: innermost runtime complexity wrt. defined symbols {head#,if#,inc#,isEmpty#,isZero#,p#,sum#,sumList# ,tail#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE