WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__terms(X)) -> terms(X) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(dbl(X)) -> X half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) terms(N) -> cons(recip(sqr(N)),n__terms(s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,dbl,first,half,sqr ,terms} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) add#(0(),X) -> c_4() add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(0()) -> c_6() dbl#(s(X)) -> c_7(dbl#(X)) first#(X1,X2) -> c_8() first#(0(),X) -> c_9() first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) half#(0()) -> c_11() half#(dbl(X)) -> c_12() half#(s(0())) -> c_13() half#(s(s(X))) -> c_14(half#(X)) sqr#(0()) -> c_15() sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_17(sqr#(N)) terms#(X) -> c_18() Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) add#(0(),X) -> c_4() add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(0()) -> c_6() dbl#(s(X)) -> c_7(dbl#(X)) first#(X1,X2) -> c_8() first#(0(),X) -> c_9() first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) half#(0()) -> c_11() half#(dbl(X)) -> c_12() half#(s(0())) -> c_13() half#(s(s(X))) -> c_14(half#(X)) sqr#(0()) -> c_15() sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_17(sqr#(N)) terms#(X) -> c_18() - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(X1,X2) activate(n__terms(X)) -> terms(X) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) half(0()) -> 0() half(dbl(X)) -> X half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) terms(N) -> cons(recip(sqr(N)),n__terms(s(N))) terms(X) -> n__terms(X) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) add#(0(),X) -> c_4() add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(0()) -> c_6() dbl#(s(X)) -> c_7(dbl#(X)) first#(X1,X2) -> c_8() first#(0(),X) -> c_9() first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) half#(0()) -> c_11() half#(dbl(X)) -> c_12() half#(s(0())) -> c_13() half#(s(s(X))) -> c_14(half#(X)) sqr#(0()) -> c_15() sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_17(sqr#(N)) terms#(X) -> c_18() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) add#(0(),X) -> c_4() add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(0()) -> c_6() dbl#(s(X)) -> c_7(dbl#(X)) first#(X1,X2) -> c_8() first#(0(),X) -> c_9() first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) half#(0()) -> c_11() half#(dbl(X)) -> c_12() half#(s(0())) -> c_13() half#(s(s(X))) -> c_14(half#(X)) sqr#(0()) -> c_15() sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_17(sqr#(N)) terms#(X) -> c_18() - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,6,8,9,11,12,13,15,18} by application of Pre({1,4,6,8,9,11,12,13,15,18}) = {2,3,5,7,10,14,16,17}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) 3: activate#(n__terms(X)) -> c_3(terms#(X)) 4: add#(0(),X) -> c_4() 5: add#(s(X),Y) -> c_5(add#(X,Y)) 6: dbl#(0()) -> c_6() 7: dbl#(s(X)) -> c_7(dbl#(X)) 8: first#(X1,X2) -> c_8() 9: first#(0(),X) -> c_9() 10: first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) 11: half#(0()) -> c_11() 12: half#(dbl(X)) -> c_12() 13: half#(s(0())) -> c_13() 14: half#(s(s(X))) -> c_14(half#(X)) 15: sqr#(0()) -> c_15() 16: sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) 17: terms#(N) -> c_17(sqr#(N)) 18: terms#(X) -> c_18() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(s(X)) -> c_7(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) half#(s(s(X))) -> c_14(half#(X)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_17(sqr#(N)) - Weak DPs: activate#(X) -> c_1() add#(0(),X) -> c_4() dbl#(0()) -> c_6() first#(X1,X2) -> c_8() first#(0(),X) -> c_9() half#(0()) -> c_11() half#(dbl(X)) -> c_12() half#(s(0())) -> c_13() sqr#(0()) -> c_15() terms#(X) -> c_18() - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) -->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):5 -->_1 first#(0(),X) -> c_9():13 -->_1 first#(X1,X2) -> c_8():12 2:S:activate#(n__terms(X)) -> c_3(terms#(X)) -->_1 terms#(N) -> c_17(sqr#(N)):8 -->_1 terms#(X) -> c_18():18 3:S:add#(s(X),Y) -> c_5(add#(X,Y)) -->_1 add#(0(),X) -> c_4():10 -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3 4:S:dbl#(s(X)) -> c_7(dbl#(X)) -->_1 dbl#(0()) -> c_6():11 -->_1 dbl#(s(X)) -> c_7(dbl#(X)):4 5:S:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) -->_1 activate#(X) -> c_1():9 -->_1 activate#(n__terms(X)) -> c_3(terms#(X)):2 -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1 6:S:half#(s(s(X))) -> c_14(half#(X)) -->_1 half#(s(0())) -> c_13():16 -->_1 half#(dbl(X)) -> c_12():15 -->_1 half#(0()) -> c_11():14 -->_1 half#(s(s(X))) -> c_14(half#(X)):6 7:S:sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_2 sqr#(0()) -> c_15():17 -->_3 dbl#(0()) -> c_6():11 -->_1 add#(0(),X) -> c_4():10 -->_2 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):7 -->_3 dbl#(s(X)) -> c_7(dbl#(X)):4 -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3 8:S:terms#(N) -> c_17(sqr#(N)) -->_1 sqr#(0()) -> c_15():17 -->_1 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):7 9:W:activate#(X) -> c_1() 10:W:add#(0(),X) -> c_4() 11:W:dbl#(0()) -> c_6() 12:W:first#(X1,X2) -> c_8() 13:W:first#(0(),X) -> c_9() 14:W:half#(0()) -> c_11() 15:W:half#(dbl(X)) -> c_12() 16:W:half#(s(0())) -> c_13() 17:W:sqr#(0()) -> c_15() 18:W:terms#(X) -> c_18() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: half#(0()) -> c_11() 15: half#(dbl(X)) -> c_12() 16: half#(s(0())) -> c_13() 12: first#(X1,X2) -> c_8() 13: first#(0(),X) -> c_9() 18: terms#(X) -> c_18() 10: add#(0(),X) -> c_4() 11: dbl#(0()) -> c_6() 17: sqr#(0()) -> c_15() 9: activate#(X) -> c_1() * Step 5: Decompose WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(s(X)) -> c_7(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) half#(s(s(X))) -> c_14(half#(X)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_17(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(s(X)) -> c_7(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_17(sqr#(N)) - Weak DPs: half#(s(s(X))) -> c_14(half#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} Problem (S) - Strict DPs: half#(s(s(X))) -> c_14(half#(X)) - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(s(X)) -> c_7(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_17(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} ** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(s(X)) -> c_7(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_17(sqr#(N)) - Weak DPs: half#(s(s(X))) -> c_14(half#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) -->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):5 2:S:activate#(n__terms(X)) -> c_3(terms#(X)) -->_1 terms#(N) -> c_17(sqr#(N)):8 3:S:add#(s(X),Y) -> c_5(add#(X,Y)) -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3 4:S:dbl#(s(X)) -> c_7(dbl#(X)) -->_1 dbl#(s(X)) -> c_7(dbl#(X)):4 5:S:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1 -->_1 activate#(n__terms(X)) -> c_3(terms#(X)):2 6:W:half#(s(s(X))) -> c_14(half#(X)) -->_1 half#(s(s(X))) -> c_14(half#(X)):6 7:S:sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3 -->_3 dbl#(s(X)) -> c_7(dbl#(X)):4 -->_2 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):7 8:S:terms#(N) -> c_17(sqr#(N)) -->_1 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: half#(s(s(X))) -> c_14(half#(X)) ** Step 5.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(s(X)) -> c_7(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_17(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) 2: activate#(n__terms(X)) -> c_3(terms#(X)) 4: dbl#(s(X)) -> c_7(dbl#(X)) 5: first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) Consider the set of all dependency pairs 1: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) 2: activate#(n__terms(X)) -> c_3(terms#(X)) 3: add#(s(X),Y) -> c_5(add#(X,Y)) 4: dbl#(s(X)) -> c_7(dbl#(X)) 5: first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) 7: sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) 8: terms#(N) -> c_17(sqr#(N)) Processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {1,2,4,5} These cover all (indirect) predecessors of dependency pairs {1,2,4,5,8} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 5.a:2.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(s(X)) -> c_7(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_17(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_7) = {1}, uargs(c_10) = {1}, uargs(c_16) = {1,2,3}, uargs(c_17) = {1} Following symbols are considered usable: {activate#,add#,dbl#,first#,half#,sqr#,terms#} TcT has computed the following interpretation: p(0) = [0] [0] [1] p(activate) = [0] [0] [0] p(add) = [0 0 0] [0] [0 0 0] x1 + [0] [1 1 0] [0] p(cons) = [1 1 1] [0] [0 0 0] x2 + [1] [0 0 1] [0] p(dbl) = [0 0 0] [0] [0 0 0] x1 + [0] [1 0 0] [0] p(first) = [0] [0] [0] p(half) = [0] [0] [0] p(n__first) = [0 1 1] [1 1 0] [0] [0 0 1] x1 + [0 0 1] x2 + [1] [0 0 0] [0 0 0] [1] p(n__terms) = [1 1 1] [1] [0 0 1] x1 + [1] [0 0 0] [1] p(nil) = [0] [0] [0] p(recip) = [0] [0] [0] p(s) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(sqr) = [0 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] p(terms) = [0] [0] [0] p(activate#) = [1 1 1] [0] [1 1 0] x1 + [0] [1 0 1] [1] p(add#) = [0] [1] [1] p(dbl#) = [0 0 1] [1] [1 0 0] x1 + [1] [0 0 0] [1] p(first#) = [0 0 0] [1 1 1] [1] [0 0 1] x1 + [1 0 0] x2 + [0] [0 1 1] [0 0 0] [0] p(half#) = [0] [0] [0] p(sqr#) = [1 1 1] [0] [1 1 0] x1 + [0] [0 0 1] [1] p(terms#) = [1 1 1] [0] [0 0 1] x1 + [0] [1 1 1] [0] p(c_1) = [0] [0] [0] p(c_2) = [1 0 1] [0] [0 0 1] x1 + [1] [0 1 0] [0] p(c_3) = [1 1 0] [0] [0 0 1] x1 + [0] [1 0 0] [1] p(c_4) = [0] [0] [0] p(c_5) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(c_6) = [0] [0] [0] p(c_7) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(c_8) = [0] [0] [0] p(c_9) = [0] [0] [0] p(c_10) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(c_11) = [0] [0] [0] p(c_12) = [0] [0] [0] p(c_13) = [0] [0] [0] p(c_14) = [0] [0] [0] p(c_15) = [0] [0] [0] p(c_16) = [1 0 0] [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [0 0 1] [0 0 0] [0 0 0] [1] p(c_17) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_18) = [0] [0] [0] Following rules are strictly oriented: activate#(n__first(X1,X2)) = [0 1 2] [1 1 1] [2] [0 1 2] X1 + [1 1 1] X2 + [1] [0 1 1] [1 1 0] [2] > [0 1 1] [1 1 1] [1] [0 1 1] X1 + [0 0 0] X2 + [1] [0 0 1] [1 0 0] [0] = c_2(first#(X1,X2)) activate#(n__terms(X)) = [1 1 2] [3] [1 1 2] X + [2] [1 1 1] [3] > [1 1 2] [0] [1 1 1] X + [0] [1 1 1] [1] = c_3(terms#(X)) dbl#(s(X)) = [0 0 1] [2] [1 1 0] X + [1] [0 0 0] [1] > [0 0 1] [1] [1 0 0] X + [1] [0 0 0] [1] = c_7(dbl#(X)) first#(s(X),cons(Y,Z)) = [0 0 0] [1 1 2] [2] [0 0 1] X + [1 1 1] Z + [1] [0 0 2] [0 0 0] [1] > [1 1 1] [0] [0 0 0] Z + [1] [0 0 0] [0] = c_10(activate#(Z)) Following rules are (at-least) weakly oriented: add#(s(X),Y) = [0] [1] [1] >= [0] [1] [0] = c_5(add#(X,Y)) sqr#(s(X)) = [1 1 2] [1] [1 1 1] X + [0] [0 0 1] [2] >= [1 1 2] [1] [0 0 0] X + [0] [0 0 0] [2] = c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) = [1 1 1] [0] [0 0 1] N + [0] [1 1 1] [0] >= [1 1 1] [0] [0 0 0] N + [0] [0 0 0] [0] = c_17(sqr#(N)) *** Step 5.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_5(add#(X,Y)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_17(sqr#(N)) - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) dbl#(s(X)) -> c_7(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_5(add#(X,Y)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) dbl#(s(X)) -> c_7(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) terms#(N) -> c_17(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:add#(s(X),Y) -> c_5(add#(X,Y)) -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):1 2:S:sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_3 dbl#(s(X)) -> c_7(dbl#(X)):5 -->_2 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2 -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):1 3:W:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) -->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):6 4:W:activate#(n__terms(X)) -> c_3(terms#(X)) -->_1 terms#(N) -> c_17(sqr#(N)):7 5:W:dbl#(s(X)) -> c_7(dbl#(X)) -->_1 dbl#(s(X)) -> c_7(dbl#(X)):5 6:W:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) -->_1 activate#(n__terms(X)) -> c_3(terms#(X)):4 -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):3 7:W:terms#(N) -> c_17(sqr#(N)) -->_1 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: dbl#(s(X)) -> c_7(dbl#(X)) *** Step 5.a:2.b:2: SimplifyRHS WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_5(add#(X,Y)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) terms#(N) -> c_17(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:add#(s(X),Y) -> c_5(add#(X,Y)) -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):1 2:S:sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_2 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2 -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):1 3:W:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) -->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):6 4:W:activate#(n__terms(X)) -> c_3(terms#(X)) -->_1 terms#(N) -> c_17(sqr#(N)):7 6:W:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) -->_1 activate#(n__terms(X)) -> c_3(terms#(X)):4 -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):3 7:W:terms#(N) -> c_17(sqr#(N)) -->_1 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)) *** Step 5.a:2.b:3: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_5(add#(X,Y)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)) - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) terms#(N) -> c_17(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)) terms#(N) -> c_17(sqr#(N)) and a lower component add#(s(X),Y) -> c_5(add#(X,Y)) Further, following extension rules are added to the lower component. activate#(n__first(X1,X2)) -> first#(X1,X2) activate#(n__terms(X)) -> terms#(X) first#(s(X),cons(Y,Z)) -> activate#(Z) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> sqr#(X) terms#(N) -> sqr#(N) **** Step 5.a:2.b:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)) - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) terms#(N) -> c_17(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)) The strictly oriented rules are moved into the weak component. ***** Step 5.a:2.b:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)) - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) terms#(N) -> c_17(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and 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}, uargs(c_3) = {1}, uargs(c_10) = {1}, uargs(c_16) = {1,2}, uargs(c_17) = {1} Following symbols are considered usable: {activate#,add#,dbl#,first#,half#,sqr#,terms#} TcT has computed the following interpretation: p(0) = [0] p(activate) = [2] x1 + [1] p(add) = [3] x1 + [5] x2 + [8] p(cons) = [1] x2 + [1] p(dbl) = [2] x1 + [2] p(first) = [1] x1 + [1] x2 + [1] p(half) = [2] p(n__first) = [1] x1 + [1] x2 + [1] p(n__terms) = [1] x1 + [0] p(nil) = [1] p(recip) = [1] p(s) = [1] x1 + [1] p(sqr) = [2] x1 + [1] p(terms) = [1] x1 + [1] p(activate#) = [8] x1 + [12] p(add#) = [0] p(dbl#) = [8] x1 + [1] p(first#) = [8] x1 + [8] x2 + [3] p(half#) = [2] x1 + [0] p(sqr#) = [4] x1 + [0] p(terms#) = [8] x1 + [1] p(c_1) = [1] p(c_2) = [1] x1 + [8] p(c_3) = [1] x1 + [10] p(c_4) = [1] p(c_5) = [1] x1 + [2] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [4] p(c_9) = [1] p(c_10) = [1] x1 + [1] p(c_11) = [1] p(c_12) = [1] p(c_13) = [1] p(c_14) = [0] p(c_15) = [0] p(c_16) = [8] x1 + [1] x2 + [2] p(c_17) = [2] x1 + [0] p(c_18) = [0] Following rules are strictly oriented: sqr#(s(X)) = [4] X + [4] > [4] X + [2] = c_16(add#(sqr(X),dbl(X)),sqr#(X)) Following rules are (at-least) weakly oriented: activate#(n__first(X1,X2)) = [8] X1 + [8] X2 + [20] >= [8] X1 + [8] X2 + [11] = c_2(first#(X1,X2)) activate#(n__terms(X)) = [8] X + [12] >= [8] X + [11] = c_3(terms#(X)) first#(s(X),cons(Y,Z)) = [8] X + [8] Z + [19] >= [8] Z + [13] = c_10(activate#(Z)) terms#(N) = [8] N + [1] >= [8] N + [0] = c_17(sqr#(N)) ***** Step 5.a:2.b:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)) terms#(N) -> c_17(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 5.a:2.b:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)) terms#(N) -> c_17(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) -->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):3 2:W:activate#(n__terms(X)) -> c_3(terms#(X)) -->_1 terms#(N) -> c_17(sqr#(N)):5 3:W:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) -->_1 activate#(n__terms(X)) -> c_3(terms#(X)):2 -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1 4:W:sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)) -->_2 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)):4 5:W:terms#(N) -> c_17(sqr#(N)) -->_1 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) 3: first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) 2: activate#(n__terms(X)) -> c_3(terms#(X)) 5: terms#(N) -> c_17(sqr#(N)) 4: sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X)) ***** Step 5.a:2.b:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 5.a:2.b:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_5(add#(X,Y)) - Weak DPs: activate#(n__first(X1,X2)) -> first#(X1,X2) activate#(n__terms(X)) -> terms#(X) first#(s(X),cons(Y,Z)) -> activate#(Z) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> sqr#(X) terms#(N) -> sqr#(N) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: add#(s(X),Y) -> c_5(add#(X,Y)) The strictly oriented rules are moved into the weak component. ***** Step 5.a:2.b:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_5(add#(X,Y)) - Weak DPs: activate#(n__first(X1,X2)) -> first#(X1,X2) activate#(n__terms(X)) -> terms#(X) first#(s(X),cons(Y,Z)) -> activate#(Z) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> sqr#(X) terms#(N) -> sqr#(N) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_5) = {1} Following symbols are considered usable: {add,dbl,sqr,activate#,add#,dbl#,first#,half#,sqr#,terms#} TcT has computed the following interpretation: p(0) = 0 p(activate) = 1 + x1 + 2*x1^2 p(add) = x1 + x2 p(cons) = 1 + x1 + x2 p(dbl) = 3*x1 p(first) = 2*x2 p(half) = 1 + x1 + x1^2 p(n__first) = x2 p(n__terms) = x1 p(nil) = 4 p(recip) = x1 p(s) = 1 + x1 p(sqr) = 2*x1^2 p(terms) = 0 p(activate#) = 6 + x1 + 4*x1^2 p(add#) = x1 + 3*x2 p(dbl#) = 2*x1 + 2*x1^2 p(first#) = 3 + 4*x2^2 p(half#) = 0 p(sqr#) = 4 + x1 + 4*x1^2 p(terms#) = 4 + x1 + 4*x1^2 p(c_1) = 1 p(c_2) = 0 p(c_3) = 1 + x1 p(c_4) = 1 p(c_5) = x1 p(c_6) = 1 p(c_7) = 0 p(c_8) = 1 p(c_9) = 1 p(c_10) = 1 p(c_11) = 1 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 4 p(c_17) = 1 p(c_18) = 0 Following rules are strictly oriented: add#(s(X),Y) = 1 + X + 3*Y > X + 3*Y = c_5(add#(X,Y)) Following rules are (at-least) weakly oriented: activate#(n__first(X1,X2)) = 6 + X2 + 4*X2^2 >= 3 + 4*X2^2 = first#(X1,X2) activate#(n__terms(X)) = 6 + X + 4*X^2 >= 4 + X + 4*X^2 = terms#(X) first#(s(X),cons(Y,Z)) = 7 + 8*Y + 8*Y*Z + 4*Y^2 + 8*Z + 4*Z^2 >= 6 + Z + 4*Z^2 = activate#(Z) sqr#(s(X)) = 9 + 9*X + 4*X^2 >= 9*X + 2*X^2 = add#(sqr(X),dbl(X)) sqr#(s(X)) = 9 + 9*X + 4*X^2 >= 4 + X + 4*X^2 = sqr#(X) terms#(N) = 4 + N + 4*N^2 >= 4 + N + 4*N^2 = sqr#(N) add(0(),X) = X >= X = X add(s(X),Y) = 1 + X + Y >= 1 + X + Y = s(add(X,Y)) dbl(0()) = 0 >= 0 = 0() dbl(s(X)) = 3 + 3*X >= 2 + 3*X = s(s(dbl(X))) sqr(0()) = 0 >= 0 = 0() sqr(s(X)) = 2 + 4*X + 2*X^2 >= 1 + 3*X + 2*X^2 = s(add(sqr(X),dbl(X))) ***** Step 5.a:2.b:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__first(X1,X2)) -> first#(X1,X2) activate#(n__terms(X)) -> terms#(X) add#(s(X),Y) -> c_5(add#(X,Y)) first#(s(X),cons(Y,Z)) -> activate#(Z) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> sqr#(X) terms#(N) -> sqr#(N) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 5.a:2.b:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__first(X1,X2)) -> first#(X1,X2) activate#(n__terms(X)) -> terms#(X) add#(s(X),Y) -> c_5(add#(X,Y)) first#(s(X),cons(Y,Z)) -> activate#(Z) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> sqr#(X) terms#(N) -> sqr#(N) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:activate#(n__first(X1,X2)) -> first#(X1,X2) -->_1 first#(s(X),cons(Y,Z)) -> activate#(Z):4 2:W:activate#(n__terms(X)) -> terms#(X) -->_1 terms#(N) -> sqr#(N):7 3:W:add#(s(X),Y) -> c_5(add#(X,Y)) -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3 4:W:first#(s(X),cons(Y,Z)) -> activate#(Z) -->_1 activate#(n__terms(X)) -> terms#(X):2 -->_1 activate#(n__first(X1,X2)) -> first#(X1,X2):1 5:W:sqr#(s(X)) -> add#(sqr(X),dbl(X)) -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3 6:W:sqr#(s(X)) -> sqr#(X) -->_1 sqr#(s(X)) -> sqr#(X):6 -->_1 sqr#(s(X)) -> add#(sqr(X),dbl(X)):5 7:W:terms#(N) -> sqr#(N) -->_1 sqr#(s(X)) -> sqr#(X):6 -->_1 sqr#(s(X)) -> add#(sqr(X),dbl(X)):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: activate#(n__first(X1,X2)) -> first#(X1,X2) 4: first#(s(X),cons(Y,Z)) -> activate#(Z) 2: activate#(n__terms(X)) -> terms#(X) 7: terms#(N) -> sqr#(N) 6: sqr#(s(X)) -> sqr#(X) 5: sqr#(s(X)) -> add#(sqr(X),dbl(X)) 3: add#(s(X),Y) -> c_5(add#(X,Y)) ***** Step 5.a:2.b:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_14(half#(X)) - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) activate#(n__terms(X)) -> c_3(terms#(X)) add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(s(X)) -> c_7(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_17(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:half#(s(s(X))) -> c_14(half#(X)) -->_1 half#(s(s(X))) -> c_14(half#(X)):1 2:W:activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) -->_1 first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)):6 3:W:activate#(n__terms(X)) -> c_3(terms#(X)) -->_1 terms#(N) -> c_17(sqr#(N)):8 4:W:add#(s(X),Y) -> c_5(add#(X,Y)) -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):4 5:W:dbl#(s(X)) -> c_7(dbl#(X)) -->_1 dbl#(s(X)) -> c_7(dbl#(X)):5 6:W:first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) -->_1 activate#(n__terms(X)) -> c_3(terms#(X)):3 -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):2 7:W:sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_2 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):7 -->_3 dbl#(s(X)) -> c_7(dbl#(X)):5 -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):4 8:W:terms#(N) -> c_17(sqr#(N)) -->_1 sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) 6: first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) 3: activate#(n__terms(X)) -> c_3(terms#(X)) 8: terms#(N) -> c_17(sqr#(N)) 7: sqr#(s(X)) -> c_16(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) 4: add#(s(X),Y) -> c_5(add#(X,Y)) 5: dbl#(s(X)) -> c_7(dbl#(X)) ** Step 5.b:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_14(half#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: half#(s(s(X))) -> c_14(half#(X)) ** Step 5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_14(half#(X)) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: half#(s(s(X))) -> c_14(half#(X)) The strictly oriented rules are moved into the weak component. *** Step 5.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_14(half#(X)) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_14) = {1} Following symbols are considered usable: {activate#,add#,dbl#,first#,half#,sqr#,terms#} TcT has computed the following interpretation: p(0) = [8] p(activate) = [8] p(add) = [1] p(cons) = [1] p(dbl) = [1] p(first) = [1] x1 + [8] x2 + [0] p(half) = [1] x1 + [0] p(n__first) = [1] x2 + [0] p(n__terms) = [1] p(nil) = [1] p(recip) = [1] x1 + [1] p(s) = [1] x1 + [2] p(sqr) = [2] x1 + [1] p(terms) = [8] x1 + [1] p(activate#) = [1] x1 + [1] p(add#) = [8] x1 + [8] p(dbl#) = [0] p(first#) = [2] x1 + [1] x2 + [1] p(half#) = [4] x1 + [0] p(sqr#) = [2] x1 + [1] p(terms#) = [1] x1 + [0] p(c_1) = [8] p(c_2) = [1] x1 + [1] p(c_3) = [1] p(c_4) = [1] p(c_5) = [2] x1 + [1] p(c_6) = [0] p(c_7) = [1] p(c_8) = [8] p(c_9) = [0] p(c_10) = [2] x1 + [2] p(c_11) = [2] p(c_12) = [2] p(c_13) = [0] p(c_14) = [1] x1 + [15] p(c_15) = [0] p(c_16) = [2] p(c_17) = [1] x1 + [1] p(c_18) = [4] Following rules are strictly oriented: half#(s(s(X))) = [4] X + [16] > [4] X + [15] = c_14(half#(X)) Following rules are (at-least) weakly oriented: *** Step 5.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: half#(s(s(X))) -> c_14(half#(X)) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: half#(s(s(X))) -> c_14(half#(X)) - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:half#(s(s(X))) -> c_14(half#(X)) -->_1 half#(s(s(X))) -> c_14(half#(X)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: half#(s(s(X))) -> c_14(half#(X)) *** Step 5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {activate/1,add/2,dbl/1,first/2,half/1,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,half#/1,sqr#/1 ,terms#/1} / {0/0,cons/2,n__first/2,n__terms/1,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/3,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,half#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))