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__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(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))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(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__first/2,n__s/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__first,n__s,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__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(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))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(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__first/2,n__s/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__first,n__s,n__terms,nil,recip} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(x){x -> n__first(x,y)} = activate(n__first(x,y)) ->^+ first(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__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(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))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(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__first/2,n__s/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__first,n__s,n__terms,nil,recip} + 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#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)) add#(0(),X) -> c_5() add#(s(X),Y) -> c_6(s#(add(X,Y)),add#(X,Y)) dbl#(0()) -> c_7() dbl#(s(X)) -> c_8(s#(s(dbl(X))),s#(dbl(X)),dbl#(X)) first#(X1,X2) -> c_9() first#(0(),X) -> c_10() first#(s(X),cons(Y,Z)) -> c_11(activate#(Z)) s#(X) -> c_12() sqr#(0()) -> c_13() sqr#(s(X)) -> c_14(s#(add(sqr(X),dbl(X))),add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_15(sqr#(N)) terms#(X) -> c_16() 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__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)) add#(0(),X) -> c_5() add#(s(X),Y) -> c_6(s#(add(X,Y)),add#(X,Y)) dbl#(0()) -> c_7() dbl#(s(X)) -> c_8(s#(s(dbl(X))),s#(dbl(X)),dbl#(X)) first#(X1,X2) -> c_9() first#(0(),X) -> c_10() first#(s(X),cons(Y,Z)) -> c_11(activate#(Z)) s#(X) -> c_12() sqr#(0()) -> c_13() sqr#(s(X)) -> c_14(s#(add(sqr(X),dbl(X))),add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_15(sqr#(N)) terms#(X) -> c_16() - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(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))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(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__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/3,c_3/2,c_4/2,c_5/0,c_6/2 ,c_7/0,c_8/3,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,n__terms,nil,recip} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,5,6,7,8,9,10,11,12,13,14,16} by application of Pre({1,5,6,7,8,9,10,11,12,13,14,16}) = {2,3,4,15}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 3: activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) 4: activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)) 5: add#(0(),X) -> c_5() 6: add#(s(X),Y) -> c_6(s#(add(X,Y)),add#(X,Y)) 7: dbl#(0()) -> c_7() 8: dbl#(s(X)) -> c_8(s#(s(dbl(X))),s#(dbl(X)),dbl#(X)) 9: first#(X1,X2) -> c_9() 10: first#(0(),X) -> c_10() 11: first#(s(X),cons(Y,Z)) -> c_11(activate#(Z)) 12: s#(X) -> c_12() 13: sqr#(0()) -> c_13() 14: sqr#(s(X)) -> c_14(s#(add(sqr(X),dbl(X))),add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) 15: terms#(N) -> c_15(sqr#(N)) 16: terms#(X) -> c_16() ** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)) terms#(N) -> c_15(sqr#(N)) - Weak DPs: activate#(X) -> c_1() add#(0(),X) -> c_5() add#(s(X),Y) -> c_6(s#(add(X,Y)),add#(X,Y)) dbl#(0()) -> c_7() dbl#(s(X)) -> c_8(s#(s(dbl(X))),s#(dbl(X)),dbl#(X)) first#(X1,X2) -> c_9() first#(0(),X) -> c_10() first#(s(X),cons(Y,Z)) -> c_11(activate#(Z)) s#(X) -> c_12() sqr#(0()) -> c_13() sqr#(s(X)) -> c_14(s#(add(sqr(X),dbl(X))),add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(X) -> c_16() - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(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))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(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__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/3,c_3/2,c_4/2,c_5/0,c_6/2 ,c_7/0,c_8/3,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,n__terms,nil,recip} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4} by application of Pre({4}) = {3}. Here rules are labelled as follows: 1: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 2: activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) 3: activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)) 4: terms#(N) -> c_15(sqr#(N)) 5: activate#(X) -> c_1() 6: add#(0(),X) -> c_5() 7: add#(s(X),Y) -> c_6(s#(add(X,Y)),add#(X,Y)) 8: dbl#(0()) -> c_7() 9: dbl#(s(X)) -> c_8(s#(s(dbl(X))),s#(dbl(X)),dbl#(X)) 10: first#(X1,X2) -> c_9() 11: first#(0(),X) -> c_10() 12: first#(s(X),cons(Y,Z)) -> c_11(activate#(Z)) 13: s#(X) -> c_12() 14: sqr#(0()) -> c_13() 15: sqr#(s(X)) -> c_14(s#(add(sqr(X),dbl(X))),add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) 16: terms#(X) -> c_16() ** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)) - Weak DPs: activate#(X) -> c_1() add#(0(),X) -> c_5() add#(s(X),Y) -> c_6(s#(add(X,Y)),add#(X,Y)) dbl#(0()) -> c_7() dbl#(s(X)) -> c_8(s#(s(dbl(X))),s#(dbl(X)),dbl#(X)) first#(X1,X2) -> c_9() first#(0(),X) -> c_10() first#(s(X),cons(Y,Z)) -> c_11(activate#(Z)) s#(X) -> c_12() sqr#(0()) -> c_13() sqr#(s(X)) -> c_14(s#(add(sqr(X),dbl(X))),add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_15(sqr#(N)) terms#(X) -> c_16() - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(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))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(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__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/3,c_3/2,c_4/2,c_5/0,c_6/2 ,c_7/0,c_8/3,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,n__terms,nil,recip} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)):3 -->_2 activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)):3 -->_3 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_1 first#(0(),X) -> c_10():10 -->_1 first#(X1,X2) -> c_9():9 -->_3 activate#(X) -> c_1():4 -->_2 activate#(X) -> c_1():4 -->_3 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 -->_2 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 2:S:activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) -->_2 activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)):3 -->_1 s#(X) -> c_12():12 -->_2 activate#(X) -> c_1():4 -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_2 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 3:S:activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)) -->_1 terms#(N) -> c_15(sqr#(N)):15 -->_1 terms#(X) -> c_16():16 -->_2 activate#(X) -> c_1():4 -->_2 activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)):3 -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_2 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 4:W:activate#(X) -> c_1() 5:W:add#(0(),X) -> c_5() 6:W:add#(s(X),Y) -> c_6(s#(add(X,Y)),add#(X,Y)) 7:W:dbl#(0()) -> c_7() 8:W:dbl#(s(X)) -> c_8(s#(s(dbl(X))),s#(dbl(X)),dbl#(X)) 9:W:first#(X1,X2) -> c_9() 10:W:first#(0(),X) -> c_10() 11:W:first#(s(X),cons(Y,Z)) -> c_11(activate#(Z)) 12:W:s#(X) -> c_12() 13:W:sqr#(0()) -> c_13() 14:W:sqr#(s(X)) -> c_14(s#(add(sqr(X),dbl(X))),add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) 15:W:terms#(N) -> c_15(sqr#(N)) -->_1 sqr#(0()) -> c_13():13 16:W:terms#(X) -> c_16() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: sqr#(s(X)) -> c_14(s#(add(sqr(X),dbl(X))),add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) 11: first#(s(X),cons(Y,Z)) -> c_11(activate#(Z)) 8: dbl#(s(X)) -> c_8(s#(s(dbl(X))),s#(dbl(X)),dbl#(X)) 7: dbl#(0()) -> c_7() 6: add#(s(X),Y) -> c_6(s#(add(X,Y)),add#(X,Y)) 5: add#(0(),X) -> c_5() 9: first#(X1,X2) -> c_9() 10: first#(0(),X) -> c_10() 12: s#(X) -> c_12() 4: activate#(X) -> c_1() 16: terms#(X) -> c_16() 15: terms#(N) -> c_15(sqr#(N)) 13: sqr#(0()) -> c_13() ** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(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))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(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__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/3,c_3/2,c_4/2,c_5/0,c_6/2 ,c_7/0,c_8/3,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,n__terms,nil,recip} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)):3 -->_2 activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)):3 -->_3 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_3 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 -->_2 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 2:S:activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) -->_2 activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)):3 -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_2 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 3:S:activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)) -->_2 activate#(n__terms(X)) -> c_4(terms#(activate(X)),activate#(X)):3 -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_2 activate#(n__first(X1,X2)) -> c_2(first#(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__first(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_3(activate#(X)) activate#(n__terms(X)) -> c_4(activate#(X)) ** Step 1.b:6: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_3(activate#(X)) activate#(n__terms(X)) -> c_4(activate#(X)) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(n__terms(X)) -> terms(activate(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))) s(X) -> n__s(X) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(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__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0,c_6/2 ,c_7/0,c_8/3,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,n__terms,nil,recip} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(n__first(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_3(activate#(X)) activate#(n__terms(X)) -> c_4(activate#(X)) ** Step 1.b:7: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_3(activate#(X)) activate#(n__terms(X)) -> c_4(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__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0,c_6/2 ,c_7/0,c_8/3,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,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} 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__first) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [0] p(n__terms) = [1] x1 + [3] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [0] p(sqr) = [0] p(terms) = [0] p(activate#) = [1] x1 + [0] 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 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] Following rules are strictly oriented: activate#(n__terms(X)) = [1] X + [3] > [1] X + [0] = c_4(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__first(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = c_2(activate#(X1),activate#(X2)) activate#(n__s(X)) = [1] X + [0] >= [1] X + [0] = c_3(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:8: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_3(activate#(X)) - Weak DPs: activate#(n__terms(X)) -> c_4(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__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0,c_6/2 ,c_7/0,c_8/3,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,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} 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__first) = [1] x1 + [1] x2 + [0] p(n__s) = [1] x1 + [1] 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#) = [1] x1 + [0] 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 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] Following rules are strictly oriented: activate#(n__s(X)) = [1] X + [1] > [1] X + [0] = c_3(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__first(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = c_2(activate#(X1),activate#(X2)) activate#(n__terms(X)) = [1] X + [0] >= [1] X + [0] = c_4(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:9: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(activate#(X1),activate#(X2)) - Weak DPs: activate#(n__s(X)) -> c_3(activate#(X)) activate#(n__terms(X)) -> c_4(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__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0,c_6/2 ,c_7/0,c_8/3,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,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} 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__first) = [1] x1 + [1] x2 + [5] p(n__s) = [1] x1 + [0] p(n__terms) = [1] x1 + [0] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [0] p(sqr) = [2] x1 + [0] p(terms) = [1] x1 + [0] p(activate#) = [1] x1 + [0] p(add#) = [4] p(dbl#) = [1] p(first#) = [1] x2 + [1] p(s#) = [8] x1 + [4] p(sqr#) = [8] p(terms#) = [1] x1 + [2] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] Following rules are strictly oriented: activate#(n__first(X1,X2)) = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [0] = c_2(activate#(X1),activate#(X2)) Following rules are (at-least) weakly oriented: activate#(n__s(X)) = [1] X + [0] >= [1] X + [0] = c_3(activate#(X)) activate#(n__terms(X)) = [1] X + [0] >= [1] X + [0] = c_4(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:10: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__first(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__s(X)) -> c_3(activate#(X)) activate#(n__terms(X)) -> c_4(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__first/2,n__s/1,n__terms/1,nil/0,recip/1,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0,c_6/2 ,c_7/0,c_8/3,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,s#,sqr# ,terms#} and constructors {0,cons,n__first,n__s,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))