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__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(X) -> n__sqr(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1 ,n__sqr/1,n__terms/1,nil/0,recip/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0 ,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + 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__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(X) -> n__sqr(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1 ,n__sqr/1,n__terms/1,nil/0,recip/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0 ,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(x){x -> n__add(x,y)} = activate(n__add(x,y)) ->^+ add(activate(x),activate(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__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(X) -> n__sqr(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1 ,n__sqr/1,n__terms/1,nil/0,recip/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,s,sqr,terms} and constructors {0 ,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)) activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_5(s#(X)) activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)) activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)) add#(X1,X2) -> c_8() add#(0(),X) -> c_9() add#(s(X),Y) -> c_10(s#(n__add(activate(X),Y)),activate#(X)) dbl#(X) -> c_11() dbl#(0()) -> c_12() dbl#(s(X)) -> c_13(s#(n__s(n__dbl(activate(X)))),activate#(X)) first#(X1,X2) -> c_14() first#(0(),X) -> c_15() first#(s(X),cons(Y,Z)) -> c_16(activate#(X),activate#(Z)) s#(X) -> c_17() sqr#(X) -> c_18() sqr#(0()) -> c_19() sqr#(s(X)) -> c_20(s#(n__add(n__sqr(activate(X)),n__dbl(activate(X)))),activate#(X),activate#(X)) terms#(N) -> c_21(sqr#(N)) terms#(X) -> c_22() 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__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)) activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_5(s#(X)) activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)) activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)) add#(X1,X2) -> c_8() add#(0(),X) -> c_9() add#(s(X),Y) -> c_10(s#(n__add(activate(X),Y)),activate#(X)) dbl#(X) -> c_11() dbl#(0()) -> c_12() dbl#(s(X)) -> c_13(s#(n__s(n__dbl(activate(X)))),activate#(X)) first#(X1,X2) -> c_14() first#(0(),X) -> c_15() first#(s(X),cons(Y,Z)) -> c_16(activate#(X),activate#(Z)) s#(X) -> c_17() sqr#(X) -> c_18() sqr#(0()) -> c_19() sqr#(s(X)) -> c_20(s#(n__add(n__sqr(activate(X)),n__dbl(activate(X)))),activate#(X),activate#(X)) terms#(N) -> c_21(sqr#(N)) terms#(X) -> c_22() - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(X) -> n__sqr(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__sqr/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/3 ,c_3/2,c_4/3,c_5/1,c_6/2,c_7/2,c_8/0,c_9/0,c_10/2,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/0,c_18/0 ,c_19/0,c_20/3,c_21/1,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,8,9,10,11,12,13,14,15,16,17,18,19,20,22} by application of Pre({1,8,9,10,11,12,13,14,15,16,17,18,19,20,22}) = {2,3,4,5,6,7,21}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 3: activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)) 4: activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 5: activate#(n__s(X)) -> c_5(s#(X)) 6: activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)) 7: activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)) 8: add#(X1,X2) -> c_8() 9: add#(0(),X) -> c_9() 10: add#(s(X),Y) -> c_10(s#(n__add(activate(X),Y)),activate#(X)) 11: dbl#(X) -> c_11() 12: dbl#(0()) -> c_12() 13: dbl#(s(X)) -> c_13(s#(n__s(n__dbl(activate(X)))),activate#(X)) 14: first#(X1,X2) -> c_14() 15: first#(0(),X) -> c_15() 16: first#(s(X),cons(Y,Z)) -> c_16(activate#(X),activate#(Z)) 17: s#(X) -> c_17() 18: sqr#(X) -> c_18() 19: sqr#(0()) -> c_19() 20: sqr#(s(X)) -> c_20(s#(n__add(n__sqr(activate(X)),n__dbl(activate(X)))),activate#(X),activate#(X)) 21: terms#(N) -> c_21(sqr#(N)) 22: terms#(X) -> c_22() ** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)) activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_5(s#(X)) activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)) activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)) terms#(N) -> c_21(sqr#(N)) - Weak DPs: activate#(X) -> c_1() add#(X1,X2) -> c_8() add#(0(),X) -> c_9() add#(s(X),Y) -> c_10(s#(n__add(activate(X),Y)),activate#(X)) dbl#(X) -> c_11() dbl#(0()) -> c_12() dbl#(s(X)) -> c_13(s#(n__s(n__dbl(activate(X)))),activate#(X)) first#(X1,X2) -> c_14() first#(0(),X) -> c_15() first#(s(X),cons(Y,Z)) -> c_16(activate#(X),activate#(Z)) s#(X) -> c_17() sqr#(X) -> c_18() sqr#(0()) -> c_19() sqr#(s(X)) -> c_20(s#(n__add(n__sqr(activate(X)),n__dbl(activate(X)))),activate#(X),activate#(X)) terms#(X) -> c_22() - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(X) -> n__sqr(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__sqr/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/3 ,c_3/2,c_4/3,c_5/1,c_6/2,c_7/2,c_8/0,c_9/0,c_10/2,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/0,c_18/0 ,c_19/0,c_20/3,c_21/1,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4,7} by application of Pre({4,7}) = {1,2,3,5,6}. Here rules are labelled as follows: 1: activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 2: activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)) 3: activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 4: activate#(n__s(X)) -> c_5(s#(X)) 5: activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)) 6: activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)) 7: terms#(N) -> c_21(sqr#(N)) 8: activate#(X) -> c_1() 9: add#(X1,X2) -> c_8() 10: add#(0(),X) -> c_9() 11: add#(s(X),Y) -> c_10(s#(n__add(activate(X),Y)),activate#(X)) 12: dbl#(X) -> c_11() 13: dbl#(0()) -> c_12() 14: dbl#(s(X)) -> c_13(s#(n__s(n__dbl(activate(X)))),activate#(X)) 15: first#(X1,X2) -> c_14() 16: first#(0(),X) -> c_15() 17: first#(s(X),cons(Y,Z)) -> c_16(activate#(X),activate#(Z)) 18: s#(X) -> c_17() 19: sqr#(X) -> c_18() 20: sqr#(0()) -> c_19() 21: sqr#(s(X)) -> c_20(s#(n__add(n__sqr(activate(X)),n__dbl(activate(X)))),activate#(X),activate#(X)) 22: terms#(X) -> c_22() ** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)) activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)) activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)) - Weak DPs: activate#(X) -> c_1() activate#(n__s(X)) -> c_5(s#(X)) add#(X1,X2) -> c_8() add#(0(),X) -> c_9() add#(s(X),Y) -> c_10(s#(n__add(activate(X),Y)),activate#(X)) dbl#(X) -> c_11() dbl#(0()) -> c_12() dbl#(s(X)) -> c_13(s#(n__s(n__dbl(activate(X)))),activate#(X)) first#(X1,X2) -> c_14() first#(0(),X) -> c_15() first#(s(X),cons(Y,Z)) -> c_16(activate#(X),activate#(Z)) s#(X) -> c_17() sqr#(X) -> c_18() sqr#(0()) -> c_19() sqr#(s(X)) -> c_20(s#(n__add(n__sqr(activate(X)),n__dbl(activate(X)))),activate#(X),activate#(X)) terms#(N) -> c_21(sqr#(N)) terms#(X) -> c_22() - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(X) -> n__sqr(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__sqr/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/3 ,c_3/2,c_4/3,c_5/1,c_6/2,c_7/2,c_8/0,c_9/0,c_10/2,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/0,c_18/0 ,c_19/0,c_20/3,c_21/1,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__s(X)) -> c_5(s#(X)):7 -->_2 activate#(n__s(X)) -> c_5(s#(X)):7 -->_3 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_2 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_3 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_2 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_3 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_2 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_3 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_2 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_1 add#(0(),X) -> c_9():9 -->_1 add#(X1,X2) -> c_8():8 -->_3 activate#(X) -> c_1():6 -->_2 activate#(X) -> c_1():6 -->_3 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 -->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 2:S:activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(X)):7 -->_2 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_2 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_2 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_1 dbl#(0()) -> c_12():12 -->_1 dbl#(X) -> c_11():11 -->_2 activate#(X) -> c_1():6 -->_2 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 3:S:activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__s(X)) -> c_5(s#(X)):7 -->_2 activate#(n__s(X)) -> c_5(s#(X)):7 -->_3 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_2 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_3 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_2 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_1 first#(0(),X) -> c_15():15 -->_1 first#(X1,X2) -> c_14():14 -->_3 activate#(X) -> c_1():6 -->_2 activate#(X) -> c_1():6 -->_3 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_2 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_3 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_2 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_3 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 -->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 4:S:activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(X)):7 -->_2 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_1 sqr#(0()) -> c_19():19 -->_1 sqr#(X) -> c_18():18 -->_2 activate#(X) -> c_1():6 -->_2 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_2 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_2 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 5:S:activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)) -->_1 terms#(N) -> c_21(sqr#(N)):21 -->_2 activate#(n__s(X)) -> c_5(s#(X)):7 -->_1 terms#(X) -> c_22():22 -->_2 activate#(X) -> c_1():6 -->_2 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_2 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_2 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_2 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 6:W:activate#(X) -> c_1() 7:W:activate#(n__s(X)) -> c_5(s#(X)) -->_1 s#(X) -> c_17():17 8:W:add#(X1,X2) -> c_8() 9:W:add#(0(),X) -> c_9() 10:W:add#(s(X),Y) -> c_10(s#(n__add(activate(X),Y)),activate#(X)) 11:W:dbl#(X) -> c_11() 12:W:dbl#(0()) -> c_12() 13:W:dbl#(s(X)) -> c_13(s#(n__s(n__dbl(activate(X)))),activate#(X)) 14:W:first#(X1,X2) -> c_14() 15:W:first#(0(),X) -> c_15() 16:W:first#(s(X),cons(Y,Z)) -> c_16(activate#(X),activate#(Z)) 17:W:s#(X) -> c_17() 18:W:sqr#(X) -> c_18() 19:W:sqr#(0()) -> c_19() 20:W:sqr#(s(X)) -> c_20(s#(n__add(n__sqr(activate(X)),n__dbl(activate(X)))),activate#(X),activate#(X)) 21:W:terms#(N) -> c_21(sqr#(N)) -->_1 sqr#(0()) -> c_19():19 -->_1 sqr#(X) -> c_18():18 22:W:terms#(X) -> c_22() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 20: sqr#(s(X)) -> c_20(s#(n__add(n__sqr(activate(X)),n__dbl(activate(X)))),activate#(X),activate#(X)) 16: first#(s(X),cons(Y,Z)) -> c_16(activate#(X),activate#(Z)) 13: dbl#(s(X)) -> c_13(s#(n__s(n__dbl(activate(X)))),activate#(X)) 10: add#(s(X),Y) -> c_10(s#(n__add(activate(X),Y)),activate#(X)) 8: add#(X1,X2) -> c_8() 9: add#(0(),X) -> c_9() 11: dbl#(X) -> c_11() 12: dbl#(0()) -> c_12() 14: first#(X1,X2) -> c_14() 15: first#(0(),X) -> c_15() 6: activate#(X) -> c_1() 22: terms#(X) -> c_22() 21: terms#(N) -> c_21(sqr#(N)) 18: sqr#(X) -> c_18() 19: sqr#(0()) -> c_19() 7: activate#(n__s(X)) -> c_5(s#(X)) 17: s#(X) -> c_17() ** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)) activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)) activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(X) -> n__sqr(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__sqr/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/3 ,c_3/2,c_4/3,c_5/1,c_6/2,c_7/2,c_8/0,c_9/0,c_10/2,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/0,c_18/0 ,c_19/0,c_20/3,c_21/1,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_2 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_3 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_2 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_3 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_2 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_3 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_2 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_3 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 -->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 2:S:activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)) -->_2 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_2 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_2 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_2 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 3:S:activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_2 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_3 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_2 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_3 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_2 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_3 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_2 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_3 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 -->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 4:S:activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)) -->_2 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_2 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_2 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_2 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 5:S:activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)) -->_2 activate#(n__terms(X)) -> c_7(terms#(activate(X)),activate#(X)):5 -->_2 activate#(n__sqr(X)) -> c_6(sqr#(activate(X)),activate#(X)):4 -->_2 activate#(n__first(X1,X2)) -> c_4(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):3 -->_2 activate#(n__dbl(X)) -> c_3(dbl#(activate(X)),activate#(X)):2 -->_2 activate#(n__add(X1,X2)) -> c_2(add#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__add(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__dbl(X)) -> c_3(activate#(X)) activate#(n__first(X1,X2)) -> c_4(activate#(X1),activate#(X2)) activate#(n__sqr(X)) -> c_6(activate#(X)) activate#(n__terms(X)) -> c_7(activate#(X)) ** Step 1.b:6: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__add(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__dbl(X)) -> c_3(activate#(X)) activate#(n__first(X1,X2)) -> c_4(activate#(X1),activate#(X2)) activate#(n__sqr(X)) -> c_6(activate#(X)) activate#(n__terms(X)) -> c_7(activate#(X)) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__dbl(X)) -> dbl(activate(X)) activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(X) activate(n__sqr(X)) -> sqr(activate(X)) activate(n__terms(X)) -> terms(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) dbl(X) -> n__dbl(X) dbl(0()) -> 0() dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z))) s(X) -> n__s(X) sqr(X) -> n__sqr(X) sqr(0()) -> 0() sqr(s(X)) -> s(n__add(n__sqr(activate(X)),n__dbl(activate(X)))) terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__sqr/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/2 ,c_3/1,c_4/2,c_5/1,c_6/1,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/0,c_18/0 ,c_19/0,c_20/3,c_21/1,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(n__add(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__dbl(X)) -> c_3(activate#(X)) activate#(n__first(X1,X2)) -> c_4(activate#(X1),activate#(X2)) activate#(n__sqr(X)) -> c_6(activate#(X)) activate#(n__terms(X)) -> c_7(activate#(X)) ** Step 1.b:7: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__add(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__dbl(X)) -> c_3(activate#(X)) activate#(n__first(X1,X2)) -> c_4(activate#(X1),activate#(X2)) activate#(n__sqr(X)) -> c_6(activate#(X)) activate#(n__terms(X)) -> c_7(activate#(X)) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__sqr/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/2 ,c_3/1,c_4/2,c_5/1,c_6/1,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/0,c_18/0 ,c_19/0,c_20/3,c_21/1,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_3) = {1}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1} Following symbols are considered usable: {activate#,add#,dbl#,first#,s#,sqr#,terms#} TcT has computed the following interpretation: p(0) = [2] p(activate) = [1] x1 + [0] p(add) = [8] x1 + [1] x2 + [1] p(cons) = [1] x1 + [2] p(dbl) = [0] p(first) = [1] x1 + [0] p(n__add) = [1] x1 + [1] x2 + [0] p(n__dbl) = [1] x1 + [0] p(n__first) = [1] x1 + [1] x2 + [0] p(n__s) = [1] p(n__sqr) = [1] x1 + [0] p(n__terms) = [1] x1 + [2] p(nil) = [2] p(recip) = [1] x1 + [0] p(s) = [1] x1 + [0] p(sqr) = [2] x1 + [1] p(terms) = [4] x1 + [1] p(activate#) = [8] x1 + [0] p(add#) = [8] x1 + [1] x2 + [4] p(dbl#) = [2] x1 + [1] p(first#) = [1] x1 + [1] x2 + [1] p(s#) = [1] x1 + [1] p(sqr#) = [1] x1 + [0] p(terms#) = [8] x1 + [1] p(c_1) = [1] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [1] x1 + [4] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [7] p(c_8) = [0] p(c_9) = [0] p(c_10) = [8] x1 + [1] x2 + [2] p(c_11) = [1] p(c_12) = [1] p(c_13) = [0] p(c_14) = [2] p(c_15) = [8] p(c_16) = [4] x1 + [1] x2 + [1] p(c_17) = [4] p(c_18) = [8] p(c_19) = [1] p(c_20) = [1] x1 + [8] x3 + [0] p(c_21) = [1] p(c_22) = [1] Following rules are strictly oriented: activate#(n__terms(X)) = [8] X + [16] > [8] X + [7] = c_7(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__add(X1,X2)) = [8] X1 + [8] X2 + [0] >= [8] X1 + [8] X2 + [0] = c_2(activate#(X1),activate#(X2)) activate#(n__dbl(X)) = [8] X + [0] >= [8] X + [0] = c_3(activate#(X)) activate#(n__first(X1,X2)) = [8] X1 + [8] X2 + [0] >= [8] X1 + [8] X2 + [0] = c_4(activate#(X1),activate#(X2)) activate#(n__sqr(X)) = [8] X + [0] >= [8] X + [0] = c_6(activate#(X)) ** Step 1.b:8: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__add(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__dbl(X)) -> c_3(activate#(X)) activate#(n__first(X1,X2)) -> c_4(activate#(X1),activate#(X2)) activate#(n__sqr(X)) -> c_6(activate#(X)) - Weak DPs: activate#(n__terms(X)) -> c_7(activate#(X)) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__sqr/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/2 ,c_3/1,c_4/2,c_5/1,c_6/1,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/0,c_18/0 ,c_19/0,c_20/3,c_21/1,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_3) = {1}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1} Following symbols are considered usable: {activate#,add#,dbl#,first#,s#,sqr#,terms#} TcT has computed the following interpretation: p(0) = [1] p(activate) = [1] p(add) = [1] x2 + [1] p(cons) = [1] x1 + [0] p(dbl) = [8] x1 + [0] p(first) = [1] x1 + [1] x2 + [0] p(n__add) = [1] x1 + [1] x2 + [0] p(n__dbl) = [1] x1 + [0] p(n__first) = [1] x1 + [1] x2 + [4] p(n__s) = [1] x1 + [1] p(n__sqr) = [1] x1 + [0] p(n__terms) = [1] x1 + [10] p(nil) = [1] p(recip) = [1] x1 + [1] p(s) = [2] x1 + [0] p(sqr) = [1] p(terms) = [2] x1 + [1] p(activate#) = [1] x1 + [0] p(add#) = [1] x1 + [2] x2 + [8] p(dbl#) = [1] p(first#) = [1] x1 + [1] x2 + [4] p(s#) = [1] p(sqr#) = [1] x1 + [0] p(terms#) = [1] x1 + [1] p(c_1) = [8] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [1] x1 + [2] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [1] p(c_8) = [0] p(c_9) = [1] p(c_10) = [1] x1 + [8] x2 + [2] p(c_11) = [2] p(c_12) = [1] p(c_13) = [2] x2 + [0] p(c_14) = [0] p(c_15) = [1] p(c_16) = [1] p(c_17) = [1] p(c_18) = [0] p(c_19) = [8] p(c_20) = [1] x3 + [2] p(c_21) = [4] p(c_22) = [1] Following rules are strictly oriented: activate#(n__first(X1,X2)) = [1] X1 + [1] X2 + [4] > [1] X1 + [1] X2 + [0] = c_4(activate#(X1),activate#(X2)) Following rules are (at-least) weakly oriented: activate#(n__add(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = c_2(activate#(X1),activate#(X2)) activate#(n__dbl(X)) = [1] X + [0] >= [1] X + [0] = c_3(activate#(X)) activate#(n__sqr(X)) = [1] X + [0] >= [1] X + [0] = c_6(activate#(X)) activate#(n__terms(X)) = [1] X + [10] >= [1] X + [1] = c_7(activate#(X)) ** Step 1.b:9: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__add(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__dbl(X)) -> c_3(activate#(X)) activate#(n__sqr(X)) -> c_6(activate#(X)) - Weak DPs: activate#(n__first(X1,X2)) -> c_4(activate#(X1),activate#(X2)) activate#(n__terms(X)) -> c_7(activate#(X)) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__sqr/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/2 ,c_3/1,c_4/2,c_5/1,c_6/1,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/0,c_18/0 ,c_19/0,c_20/3,c_21/1,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + 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) = {1,2}, uargs(c_3) = {1}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [0] p(add) = [0] p(cons) = [1] x1 + [1] x2 + [0] p(dbl) = [0] p(first) = [0] p(n__add) = [1] x1 + [1] x2 + [0] p(n__dbl) = [1] x1 + [0] p(n__first) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [0] p(n__sqr) = [1] x1 + [7] p(n__terms) = [1] x1 + [0] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [0] p(sqr) = [0] p(terms) = [0] p(activate#) = [3] x1 + [0] p(add#) = [0] p(dbl#) = [0] p(first#) = [1] x2 + [0] p(s#) = [0] p(sqr#) = [0] p(terms#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [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] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] Following rules are strictly oriented: activate#(n__sqr(X)) = [3] X + [21] > [3] X + [0] = c_6(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__add(X1,X2)) = [3] X1 + [3] X2 + [0] >= [3] X1 + [3] X2 + [0] = c_2(activate#(X1),activate#(X2)) activate#(n__dbl(X)) = [3] X + [0] >= [3] X + [0] = c_3(activate#(X)) activate#(n__first(X1,X2)) = [3] X1 + [3] X2 + [0] >= [3] X1 + [3] X2 + [0] = c_4(activate#(X1),activate#(X2)) activate#(n__terms(X)) = [3] X + [0] >= [3] X + [0] = c_7(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:10: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__add(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__dbl(X)) -> c_3(activate#(X)) - Weak DPs: activate#(n__first(X1,X2)) -> c_4(activate#(X1),activate#(X2)) activate#(n__sqr(X)) -> c_6(activate#(X)) activate#(n__terms(X)) -> c_7(activate#(X)) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__sqr/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/2 ,c_3/1,c_4/2,c_5/1,c_6/1,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/0,c_18/0 ,c_19/0,c_20/3,c_21/1,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_3) = {1}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1} Following symbols are considered usable: {activate#,add#,dbl#,first#,s#,sqr#,terms#} TcT has computed the following interpretation: p(0) = [8] p(activate) = [0] p(add) = [2] x2 + [0] p(cons) = [1] x1 + [2] p(dbl) = [1] x1 + [1] p(first) = [4] x1 + [2] x2 + [0] p(n__add) = [1] x1 + [1] x2 + [0] p(n__dbl) = [1] x1 + [1] p(n__first) = [1] x1 + [1] x2 + [4] p(n__s) = [0] p(n__sqr) = [1] x1 + [0] p(n__terms) = [1] x1 + [0] p(nil) = [8] p(recip) = [1] x1 + [0] p(s) = [1] x1 + [0] p(sqr) = [2] x1 + [1] p(terms) = [1] x1 + [0] p(activate#) = [4] x1 + [0] p(add#) = [2] x1 + [0] p(dbl#) = [2] x1 + [1] p(first#) = [1] x1 + [2] x2 + [0] p(s#) = [0] p(sqr#) = [2] x1 + [1] p(terms#) = [2] p(c_1) = [1] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [2] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [8] p(c_10) = [1] x1 + [1] x2 + [8] p(c_11) = [1] p(c_12) = [0] p(c_13) = [2] p(c_14) = [8] p(c_15) = [0] p(c_16) = [1] x1 + [1] x2 + [0] p(c_17) = [1] p(c_18) = [0] p(c_19) = [1] p(c_20) = [1] x1 + [1] x3 + [1] p(c_21) = [1] p(c_22) = [2] Following rules are strictly oriented: activate#(n__dbl(X)) = [4] X + [4] > [4] X + [0] = c_3(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__add(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = c_2(activate#(X1),activate#(X2)) activate#(n__first(X1,X2)) = [4] X1 + [4] X2 + [16] >= [4] X1 + [4] X2 + [0] = c_4(activate#(X1),activate#(X2)) activate#(n__sqr(X)) = [4] X + [0] >= [4] X + [0] = c_6(activate#(X)) activate#(n__terms(X)) = [4] X + [0] >= [4] X + [0] = c_7(activate#(X)) ** Step 1.b:11: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__add(X1,X2)) -> c_2(activate#(X1),activate#(X2)) - Weak DPs: activate#(n__dbl(X)) -> c_3(activate#(X)) activate#(n__first(X1,X2)) -> c_4(activate#(X1),activate#(X2)) activate#(n__sqr(X)) -> c_6(activate#(X)) activate#(n__terms(X)) -> c_7(activate#(X)) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__sqr/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/2 ,c_3/1,c_4/2,c_5/1,c_6/1,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/0,c_18/0 ,c_19/0,c_20/3,c_21/1,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + 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) = {1,2}, uargs(c_3) = {1}, uargs(c_4) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [0] p(add) = [2] x1 + [8] x2 + [0] p(cons) = [0] p(dbl) = [2] p(first) = [2] x1 + [1] x2 + [2] p(n__add) = [1] x1 + [1] x2 + [5] p(n__dbl) = [1] x1 + [0] p(n__first) = [1] x1 + [1] x2 + [5] p(n__s) = [1] p(n__sqr) = [1] x1 + [9] p(n__terms) = [1] x1 + [0] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [0] p(sqr) = [0] p(terms) = [0] p(activate#) = [2] x1 + [9] p(add#) = [0] p(dbl#) = [0] p(first#) = [0] p(s#) = [0] p(sqr#) = [0] p(terms#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [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] p(c_20) = [0] p(c_21) = [8] x1 + [0] p(c_22) = [0] Following rules are strictly oriented: activate#(n__add(X1,X2)) = [2] X1 + [2] X2 + [19] > [2] X1 + [2] X2 + [18] = c_2(activate#(X1),activate#(X2)) Following rules are (at-least) weakly oriented: activate#(n__dbl(X)) = [2] X + [9] >= [2] X + [9] = c_3(activate#(X)) activate#(n__first(X1,X2)) = [2] X1 + [2] X2 + [19] >= [2] X1 + [2] X2 + [18] = c_4(activate#(X1),activate#(X2)) activate#(n__sqr(X)) = [2] X + [27] >= [2] X + [9] = c_6(activate#(X)) activate#(n__terms(X)) = [2] X + [9] >= [2] X + [9] = c_7(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:12: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__add(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__dbl(X)) -> c_3(activate#(X)) activate#(n__first(X1,X2)) -> c_4(activate#(X1),activate#(X2)) activate#(n__sqr(X)) -> c_6(activate#(X)) activate#(n__terms(X)) -> c_7(activate#(X)) - Signature: {activate/1,add/2,dbl/1,first/2,s/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,s#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__add/2,n__dbl/1,n__first/2,n__s/1,n__sqr/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/2 ,c_3/1,c_4/2,c_5/1,c_6/1,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/0,c_18/0 ,c_19/0,c_20/3,c_21/1,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__add,n__dbl,n__first,n__s,n__sqr,n__terms,nil,recip} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))