WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,Y)) -> X if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) tail(cons(X,Y)) -> activate(Y) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0 ,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,s,sieve ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,Y)) -> X if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) tail(cons(X,Y)) -> activate(Y) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0 ,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,s,sieve ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(x){x -> n__cons(x,y)} = activate(n__cons(x,y)) ->^+ cons(activate(x),y) = C[activate(x) = activate(x){}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,Y)) -> X if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) tail(cons(X,Y)) -> activate(Y) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1} / {0/0,divides/2,false/0 ,n__cons/2,n__filter/2,n__from/1,n__s/1,n__sieve/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,cons,filter,from,head,if,primes,s,sieve ,tail} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)) cons#(X1,X2) -> c_7() filter#(X1,X2) -> c_8() filter#(s(s(X)),cons(Y,Z)) -> c_9(if#(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) ,s#(s(X)) ,s#(X) ,activate#(Z)) from#(X) -> c_10(cons#(X,n__from(n__s(X)))) from#(X) -> c_11() head#(cons(X,Y)) -> c_12() if#(false(),X,Y) -> c_13(activate#(Y)) if#(true(),X,Y) -> c_14(activate#(X)) primes#() -> c_15(sieve#(from(s(s(0())))),from#(s(s(0()))),s#(s(0())),s#(0())) s#(X) -> c_16() sieve#(X) -> c_17() sieve#(cons(X,Y)) -> c_18(cons#(X,n__filter(X,n__sieve(activate(Y)))),activate#(Y)) tail#(cons(X,Y)) -> c_19(activate#(Y)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)) cons#(X1,X2) -> c_7() filter#(X1,X2) -> c_8() filter#(s(s(X)),cons(Y,Z)) -> c_9(if#(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) ,s#(s(X)) ,s#(X) ,activate#(Z)) from#(X) -> c_10(cons#(X,n__from(n__s(X)))) from#(X) -> c_11() head#(cons(X,Y)) -> c_12() if#(false(),X,Y) -> c_13(activate#(Y)) if#(true(),X,Y) -> c_14(activate#(X)) primes#() -> c_15(sieve#(from(s(s(0())))),from#(s(s(0()))),s#(s(0())),s#(0())) s#(X) -> c_16() sieve#(X) -> c_17() sieve#(cons(X,Y)) -> c_18(cons#(X,n__filter(X,n__sieve(activate(Y)))),activate#(Y)) tail#(cons(X,Y)) -> c_19(activate#(Y)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,Y)) -> X if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) tail(cons(X,Y)) -> activate(Y) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/2,c_3/3,c_4/2,c_5/2,c_6/2,c_7/0,c_8/0,c_9/4,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/4,c_16/0,c_17/0,c_18/2,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,7,8,9,11,12,16,17,18,19} by application of Pre({1,7,8,9,11,12,16,17,18,19}) = {2,3,4,5,6,10,13,14,15}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) 3: activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 4: activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) 5: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) 6: activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)) 7: cons#(X1,X2) -> c_7() 8: filter#(X1,X2) -> c_8() 9: filter#(s(s(X)),cons(Y,Z)) -> c_9(if#(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) ,s#(s(X)) ,s#(X) ,activate#(Z)) 10: from#(X) -> c_10(cons#(X,n__from(n__s(X)))) 11: from#(X) -> c_11() 12: head#(cons(X,Y)) -> c_12() 13: if#(false(),X,Y) -> c_13(activate#(Y)) 14: if#(true(),X,Y) -> c_14(activate#(X)) 15: primes#() -> c_15(sieve#(from(s(s(0())))),from#(s(s(0()))),s#(s(0())),s#(0())) 16: s#(X) -> c_16() 17: sieve#(X) -> c_17() 18: sieve#(cons(X,Y)) -> c_18(cons#(X,n__filter(X,n__sieve(activate(Y)))),activate#(Y)) 19: tail#(cons(X,Y)) -> c_19(activate#(Y)) ** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)) from#(X) -> c_10(cons#(X,n__from(n__s(X)))) if#(false(),X,Y) -> c_13(activate#(Y)) if#(true(),X,Y) -> c_14(activate#(X)) primes#() -> c_15(sieve#(from(s(s(0())))),from#(s(s(0()))),s#(s(0())),s#(0())) - Weak DPs: activate#(X) -> c_1() cons#(X1,X2) -> c_7() filter#(X1,X2) -> c_8() filter#(s(s(X)),cons(Y,Z)) -> c_9(if#(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) ,s#(s(X)) ,s#(X) ,activate#(Z)) from#(X) -> c_11() head#(cons(X,Y)) -> c_12() s#(X) -> c_16() sieve#(X) -> c_17() sieve#(cons(X,Y)) -> c_18(cons#(X,n__filter(X,n__sieve(activate(Y)))),activate#(Y)) tail#(cons(X,Y)) -> c_19(activate#(Y)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,Y)) -> X if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) tail(cons(X,Y)) -> activate(Y) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/2,c_3/3,c_4/2,c_5/2,c_6/2,c_7/0,c_8/0,c_9/4,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/4,c_16/0,c_17/0,c_18/2,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {6} by application of Pre({6}) = {3,9}. Here rules are labelled as follows: 1: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) 2: activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 3: activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) 4: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) 5: activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)) 6: from#(X) -> c_10(cons#(X,n__from(n__s(X)))) 7: if#(false(),X,Y) -> c_13(activate#(Y)) 8: if#(true(),X,Y) -> c_14(activate#(X)) 9: primes#() -> c_15(sieve#(from(s(s(0())))),from#(s(s(0()))),s#(s(0())),s#(0())) 10: activate#(X) -> c_1() 11: cons#(X1,X2) -> c_7() 12: filter#(X1,X2) -> c_8() 13: filter#(s(s(X)),cons(Y,Z)) -> c_9(if#(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) ,s#(s(X)) ,s#(X) ,activate#(Z)) 14: from#(X) -> c_11() 15: head#(cons(X,Y)) -> c_12() 16: s#(X) -> c_16() 17: sieve#(X) -> c_17() 18: sieve#(cons(X,Y)) -> c_18(cons#(X,n__filter(X,n__sieve(activate(Y)))),activate#(Y)) 19: tail#(cons(X,Y)) -> c_19(activate#(Y)) ** Step 1.b:4: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)) if#(false(),X,Y) -> c_13(activate#(Y)) if#(true(),X,Y) -> c_14(activate#(X)) primes#() -> c_15(sieve#(from(s(s(0())))),from#(s(s(0()))),s#(s(0())),s#(0())) - Weak DPs: activate#(X) -> c_1() cons#(X1,X2) -> c_7() filter#(X1,X2) -> c_8() filter#(s(s(X)),cons(Y,Z)) -> c_9(if#(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) ,s#(s(X)) ,s#(X) ,activate#(Z)) from#(X) -> c_10(cons#(X,n__from(n__s(X)))) from#(X) -> c_11() head#(cons(X,Y)) -> c_12() s#(X) -> c_16() sieve#(X) -> c_17() sieve#(cons(X,Y)) -> c_18(cons#(X,n__filter(X,n__sieve(activate(Y)))),activate#(Y)) tail#(cons(X,Y)) -> c_19(activate#(Y)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,Y)) -> X if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) tail(cons(X,Y)) -> activate(Y) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/2,c_3/3,c_4/2,c_5/2,c_6/2,c_7/0,c_8/0,c_9/4,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/4,c_16/0,c_17/0,c_18/2,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {8} by application of Pre({8}) = {}. Here rules are labelled as follows: 1: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) 2: activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 3: activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) 4: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) 5: activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)) 6: if#(false(),X,Y) -> c_13(activate#(Y)) 7: if#(true(),X,Y) -> c_14(activate#(X)) 8: primes#() -> c_15(sieve#(from(s(s(0())))),from#(s(s(0()))),s#(s(0())),s#(0())) 9: activate#(X) -> c_1() 10: cons#(X1,X2) -> c_7() 11: filter#(X1,X2) -> c_8() 12: filter#(s(s(X)),cons(Y,Z)) -> c_9(if#(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) ,s#(s(X)) ,s#(X) ,activate#(Z)) 13: from#(X) -> c_10(cons#(X,n__from(n__s(X)))) 14: from#(X) -> c_11() 15: head#(cons(X,Y)) -> c_12() 16: s#(X) -> c_16() 17: sieve#(X) -> c_17() 18: sieve#(cons(X,Y)) -> c_18(cons#(X,n__filter(X,n__sieve(activate(Y)))),activate#(Y)) 19: tail#(cons(X,Y)) -> c_19(activate#(Y)) ** Step 1.b:5: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)) if#(false(),X,Y) -> c_13(activate#(Y)) if#(true(),X,Y) -> c_14(activate#(X)) - Weak DPs: activate#(X) -> c_1() cons#(X1,X2) -> c_7() filter#(X1,X2) -> c_8() filter#(s(s(X)),cons(Y,Z)) -> c_9(if#(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) ,s#(s(X)) ,s#(X) ,activate#(Z)) from#(X) -> c_10(cons#(X,n__from(n__s(X)))) from#(X) -> c_11() head#(cons(X,Y)) -> c_12() primes#() -> c_15(sieve#(from(s(s(0())))),from#(s(s(0()))),s#(s(0())),s#(0())) s#(X) -> c_16() sieve#(X) -> c_17() sieve#(cons(X,Y)) -> c_18(cons#(X,n__filter(X,n__sieve(activate(Y)))),activate#(Y)) tail#(cons(X,Y)) -> c_19(activate#(Y)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,Y)) -> X if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) tail(cons(X,Y)) -> activate(Y) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/2,c_3/3,c_4/2,c_5/2,c_6/2,c_7/0,c_8/0,c_9/4,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/4,c_16/0,c_17/0,c_18/2,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) -->_2 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_2 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_1 cons#(X1,X2) -> c_7():9 -->_2 activate#(X) -> c_1():8 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 2:S:activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_2 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_3 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_3 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_1 filter#(X1,X2) -> c_8():10 -->_3 activate#(X) -> c_1():8 -->_2 activate#(X) -> c_1():8 -->_3 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_2 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_3 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 3:S:activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) -->_1 from#(X) -> c_10(cons#(X,n__from(n__s(X)))):12 -->_2 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_1 from#(X) -> c_11():13 -->_2 activate#(X) -> c_1():8 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_2 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 4:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) -->_2 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_1 s#(X) -> c_16():16 -->_2 activate#(X) -> c_1():8 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_2 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 5:S:activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)) -->_1 sieve#(X) -> c_17():17 -->_2 activate#(X) -> c_1():8 -->_2 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_2 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 6:S:if#(false(),X,Y) -> c_13(activate#(Y)) -->_1 activate#(X) -> c_1():8 -->_1 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_1 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_1 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_1 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_1 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 7:S:if#(true(),X,Y) -> c_14(activate#(X)) -->_1 activate#(X) -> c_1():8 -->_1 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_1 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_1 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_1 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_1 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 8:W:activate#(X) -> c_1() 9:W:cons#(X1,X2) -> c_7() 10:W:filter#(X1,X2) -> c_8() 11:W:filter#(s(s(X)),cons(Y,Z)) -> c_9(if#(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) ,s#(s(X)) ,s#(X) ,activate#(Z)) 12:W:from#(X) -> c_10(cons#(X,n__from(n__s(X)))) -->_1 cons#(X1,X2) -> c_7():9 13:W:from#(X) -> c_11() 14:W:head#(cons(X,Y)) -> c_12() 15:W:primes#() -> c_15(sieve#(from(s(s(0())))),from#(s(s(0()))),s#(s(0())),s#(0())) -->_1 sieve#(X) -> c_17():17 -->_4 s#(X) -> c_16():16 -->_3 s#(X) -> c_16():16 -->_2 from#(X) -> c_11():13 -->_2 from#(X) -> c_10(cons#(X,n__from(n__s(X)))):12 16:W:s#(X) -> c_16() 17:W:sieve#(X) -> c_17() 18:W:sieve#(cons(X,Y)) -> c_18(cons#(X,n__filter(X,n__sieve(activate(Y)))),activate#(Y)) 19:W:tail#(cons(X,Y)) -> c_19(activate#(Y)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 19: tail#(cons(X,Y)) -> c_19(activate#(Y)) 18: sieve#(cons(X,Y)) -> c_18(cons#(X,n__filter(X,n__sieve(activate(Y)))),activate#(Y)) 15: primes#() -> c_15(sieve#(from(s(s(0())))),from#(s(s(0()))),s#(s(0())),s#(0())) 14: head#(cons(X,Y)) -> c_12() 11: filter#(s(s(X)),cons(Y,Z)) -> c_9(if#(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) ,s#(s(X)) ,s#(X) ,activate#(Z)) 10: filter#(X1,X2) -> c_8() 13: from#(X) -> c_11() 12: from#(X) -> c_10(cons#(X,n__from(n__s(X)))) 9: cons#(X1,X2) -> c_7() 16: s#(X) -> c_16() 8: activate#(X) -> c_1() 17: sieve#(X) -> c_17() ** Step 1.b:6: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)) if#(false(),X,Y) -> c_13(activate#(Y)) if#(true(),X,Y) -> c_14(activate#(X)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,Y)) -> X if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) tail(cons(X,Y)) -> activate(Y) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/2,c_3/3,c_4/2,c_5/2,c_6/2,c_7/0,c_8/0,c_9/4,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/4,c_16/0,c_17/0,c_18/2,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)) -->_2 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_2 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 2:S:activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_2 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_3 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_3 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_3 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_2 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_3 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 3:S:activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) -->_2 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_2 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 4:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) -->_2 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_2 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 5:S:activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)) -->_2 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_2 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 6:S:if#(false(),X,Y) -> c_13(activate#(Y)) -->_1 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_1 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_1 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_1 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_1 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 7:S:if#(true(),X,Y) -> c_14(activate#(X)) -->_1 activate#(n__sieve(X)) -> c_6(sieve#(activate(X)),activate#(X)):5 -->_1 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_1 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):3 -->_1 activate#(n__filter(X1,X2)) -> c_3(filter#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):2 -->_1 activate#(n__cons(X1,X2)) -> c_2(cons#(activate(X1),X2),activate#(X1)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) activate#(n__sieve(X)) -> c_6(activate#(X)) ** Step 1.b:7: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) activate#(n__sieve(X)) -> c_6(activate#(X)) if#(false(),X,Y) -> c_13(activate#(Y)) if#(true(),X,Y) -> c_14(activate#(X)) - Weak TRS: activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__sieve(X)) -> sieve(activate(X)) cons(X1,X2) -> n__cons(X1,X2) filter(X1,X2) -> n__filter(X1,X2) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y) ,n__filter(n__s(n__s(X)),activate(Z)) ,n__cons(Y,n__filter(X,n__sieve(Y)))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,Y)) -> X if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) primes() -> sieve(from(s(s(0())))) s(X) -> n__s(X) sieve(X) -> n__sieve(X) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) tail(cons(X,Y)) -> activate(Y) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/2,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/4,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/4,c_16/0,c_17/0,c_18/2,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) activate#(n__sieve(X)) -> c_6(activate#(X)) if#(false(),X,Y) -> c_13(activate#(Y)) if#(true(),X,Y) -> c_14(activate#(X)) ** Step 1.b:8: RemoveHeads WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) activate#(n__sieve(X)) -> c_6(activate#(X)) if#(false(),X,Y) -> c_13(activate#(Y)) if#(true(),X,Y) -> c_14(activate#(X)) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/2,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/4,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/4,c_16/0,c_17/0,c_18/2,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) -->_1 activate#(n__sieve(X)) -> c_6(activate#(X)):5 -->_1 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_1 activate#(n__from(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2 -->_1 activate#(n__cons(X1,X2)) -> c_2(activate#(X1)):1 2:S:activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)) -->_2 activate#(n__sieve(X)) -> c_6(activate#(X)):5 -->_1 activate#(n__sieve(X)) -> c_6(activate#(X)):5 -->_2 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_1 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_2 activate#(n__from(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__from(X)) -> c_4(activate#(X)):3 -->_2 activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2 -->_1 activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2 -->_2 activate#(n__cons(X1,X2)) -> c_2(activate#(X1)):1 -->_1 activate#(n__cons(X1,X2)) -> c_2(activate#(X1)):1 3:S:activate#(n__from(X)) -> c_4(activate#(X)) -->_1 activate#(n__sieve(X)) -> c_6(activate#(X)):5 -->_1 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_1 activate#(n__from(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2 -->_1 activate#(n__cons(X1,X2)) -> c_2(activate#(X1)):1 4:S:activate#(n__s(X)) -> c_5(activate#(X)) -->_1 activate#(n__sieve(X)) -> c_6(activate#(X)):5 -->_1 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_1 activate#(n__from(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2 -->_1 activate#(n__cons(X1,X2)) -> c_2(activate#(X1)):1 5:S:activate#(n__sieve(X)) -> c_6(activate#(X)) -->_1 activate#(n__sieve(X)) -> c_6(activate#(X)):5 -->_1 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_1 activate#(n__from(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2 -->_1 activate#(n__cons(X1,X2)) -> c_2(activate#(X1)):1 6:S:if#(false(),X,Y) -> c_13(activate#(Y)) -->_1 activate#(n__sieve(X)) -> c_6(activate#(X)):5 -->_1 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_1 activate#(n__from(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2 -->_1 activate#(n__cons(X1,X2)) -> c_2(activate#(X1)):1 7:S:if#(true(),X,Y) -> c_14(activate#(X)) -->_1 activate#(n__sieve(X)) -> c_6(activate#(X)):5 -->_1 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_1 activate#(n__from(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)):2 -->_1 activate#(n__cons(X1,X2)) -> c_2(activate#(X1)):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,if#(false(),X,Y) -> c_13(activate#(Y))),(7,if#(true(),X,Y) -> c_14(activate#(X)))] ** Step 1.b:9: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) activate#(n__sieve(X)) -> c_6(activate#(X)) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/2,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/4,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/4,c_16/0,c_17/0,c_18/2,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1,2}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [0] p(cons) = [0] p(divides) = [1] x1 + [1] x2 + [0] p(false) = [0] p(filter) = [0] p(from) = [0] p(head) = [0] p(if) = [0] p(n__cons) = [1] x1 + [1] x2 + [0] p(n__filter) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [2] p(n__sieve) = [1] x1 + [0] p(primes) = [0] p(s) = [0] p(sieve) = [0] p(tail) = [0] p(true) = [0] p(activate#) = [4] x1 + [0] p(cons#) = [0] p(filter#) = [0] p(from#) = [0] p(head#) = [0] p(if#) = [0] p(primes#) = [0] p(s#) = [0] p(sieve#) = [0] p(tail#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [8] p(c_3) = [1] x1 + [1] x2 + [8] p(c_4) = [1] x1 + [8] p(c_5) = [1] x1 + [4] p(c_6) = [1] x1 + [0] p(c_7) = [2] p(c_8) = [0] p(c_9) = [1] x2 + [0] p(c_10) = [8] x1 + [1] p(c_11) = [1] p(c_12) = [4] p(c_13) = [1] p(c_14) = [1] x1 + [4] p(c_15) = [1] x1 + [1] x3 + [1] x4 + [1] p(c_16) = [1] p(c_17) = [1] p(c_18) = [2] x1 + [1] x2 + [1] p(c_19) = [8] x1 + [0] Following rules are strictly oriented: activate#(n__s(X)) = [4] X + [8] > [4] X + [4] = c_5(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__cons(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [8] = c_2(activate#(X1)) activate#(n__filter(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [8] = c_3(activate#(X1),activate#(X2)) activate#(n__from(X)) = [4] X + [0] >= [4] X + [8] = c_4(activate#(X)) activate#(n__sieve(X)) = [4] X + [0] >= [4] X + [0] = c_6(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:10: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__sieve(X)) -> c_6(activate#(X)) - Weak DPs: activate#(n__s(X)) -> c_5(activate#(X)) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/2,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/4,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/4,c_16/0,c_17/0,c_18/2,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1,2}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [0] p(cons) = [0] p(divides) = [1] x1 + [1] x2 + [0] p(false) = [0] p(filter) = [0] p(from) = [0] p(head) = [0] p(if) = [0] p(n__cons) = [1] x1 + [1] x2 + [1] p(n__filter) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [5] p(n__sieve) = [1] x1 + [1] p(primes) = [0] p(s) = [0] p(sieve) = [0] p(tail) = [0] p(true) = [0] p(activate#) = [4] x1 + [1] p(cons#) = [0] p(filter#) = [0] p(from#) = [0] p(head#) = [0] p(if#) = [0] p(primes#) = [0] p(s#) = [0] p(sieve#) = [0] p(tail#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [1] x2 + [7] p(c_4) = [1] x1 + [8] p(c_5) = [1] x1 + [4] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] Following rules are strictly oriented: activate#(n__cons(X1,X2)) = [4] X1 + [4] X2 + [5] > [4] X1 + [1] = c_2(activate#(X1)) activate#(n__sieve(X)) = [4] X + [5] > [4] X + [1] = c_6(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__filter(X1,X2)) = [4] X1 + [4] X2 + [1] >= [4] X1 + [4] X2 + [9] = c_3(activate#(X1),activate#(X2)) activate#(n__from(X)) = [4] X + [1] >= [4] X + [9] = c_4(activate#(X)) activate#(n__s(X)) = [4] X + [21] >= [4] X + [5] = c_5(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:11: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(activate#(X)) - Weak DPs: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__s(X)) -> c_5(activate#(X)) activate#(n__sieve(X)) -> c_6(activate#(X)) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/2,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/4,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/4,c_16/0,c_17/0,c_18/2,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1,2}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [0] p(cons) = [0] p(divides) = [1] x1 + [1] x2 + [0] p(false) = [0] p(filter) = [0] p(from) = [0] p(head) = [0] p(if) = [0] p(n__cons) = [1] x1 + [1] x2 + [0] p(n__filter) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [7] p(n__s) = [1] x1 + [0] p(n__sieve) = [1] x1 + [0] p(primes) = [0] p(s) = [0] p(sieve) = [0] p(tail) = [0] p(true) = [0] p(activate#) = [3] x1 + [0] p(cons#) = [0] p(filter#) = [0] p(from#) = [0] p(head#) = [0] p(if#) = [0] p(primes#) = [0] p(s#) = [0] p(sieve#) = [0] p(tail#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] Following rules are strictly oriented: activate#(n__from(X)) = [3] X + [21] > [3] X + [0] = c_4(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__cons(X1,X2)) = [3] X1 + [3] X2 + [0] >= [3] X1 + [0] = c_2(activate#(X1)) activate#(n__filter(X1,X2)) = [3] X1 + [3] X2 + [0] >= [3] X1 + [3] X2 + [0] = c_3(activate#(X1),activate#(X2)) activate#(n__s(X)) = [3] X + [0] >= [3] X + [0] = c_5(activate#(X)) activate#(n__sieve(X)) = [3] X + [0] >= [3] X + [0] = c_6(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:12: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)) - Weak DPs: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) activate#(n__sieve(X)) -> c_6(activate#(X)) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/2,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/4,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/4,c_16/0,c_17/0,c_18/2,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1,2}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [0] p(cons) = [0] p(divides) = [1] x1 + [1] x2 + [0] p(false) = [0] p(filter) = [0] p(from) = [0] p(head) = [0] p(if) = [0] p(n__cons) = [1] x1 + [1] x2 + [0] p(n__filter) = [1] x1 + [1] x2 + [9] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [0] p(n__sieve) = [1] x1 + [0] p(primes) = [0] p(s) = [0] p(sieve) = [0] p(tail) = [8] x1 + [0] p(true) = [1] p(activate#) = [2] x1 + [13] p(cons#) = [0] p(filter#) = [0] p(from#) = [1] x1 + [0] p(head#) = [1] x1 + [1] p(if#) = [2] x2 + [1] x3 + [2] p(primes#) = [2] p(s#) = [1] x1 + [2] p(sieve#) = [1] p(tail#) = [1] p(c_1) = [4] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] Following rules are strictly oriented: activate#(n__filter(X1,X2)) = [2] X1 + [2] X2 + [31] > [2] X1 + [2] X2 + [26] = c_3(activate#(X1),activate#(X2)) Following rules are (at-least) weakly oriented: activate#(n__cons(X1,X2)) = [2] X1 + [2] X2 + [13] >= [2] X1 + [13] = c_2(activate#(X1)) activate#(n__from(X)) = [2] X + [13] >= [2] X + [13] = c_4(activate#(X)) activate#(n__s(X)) = [2] X + [13] >= [2] X + [13] = c_5(activate#(X)) activate#(n__sieve(X)) = [2] X + [13] >= [2] X + [13] = c_6(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:13: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__cons(X1,X2)) -> c_2(activate#(X1)) activate#(n__filter(X1,X2)) -> c_3(activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) activate#(n__sieve(X)) -> c_6(activate#(X)) - Signature: {activate/1,cons/2,filter/2,from/1,head/1,if/3,primes/0,s/1,sieve/1,tail/1,activate#/1,cons#/2,filter#/2 ,from#/1,head#/1,if#/3,primes#/0,s#/1,sieve#/1,tail#/1} / {0/0,divides/2,false/0,n__cons/2,n__filter/2 ,n__from/1,n__s/1,n__sieve/1,true/0,c_1/0,c_2/1,c_3/2,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/4,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/1,c_15/4,c_16/0,c_17/0,c_18/2,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,cons#,filter#,from#,head#,if#,primes#,s#,sieve# ,tail#} and constructors {0,divides,false,n__cons,n__filter,n__from,n__s,n__sieve,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))