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))) 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,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,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)) sqr#(0()) -> c_11() sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N)) terms#(X) -> c_14() Weak DPs and mark the set of starting terms. * Step 2: 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)) sqr#(0()) -> c_11() sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N)) terms#(X) -> c_14() - 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))) 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,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,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_12/3,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,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 {1,4,6,8,9,11,14} by application of Pre({1,4,6,8,9,11,14}) = {2,3,5,7,10,12,13}. 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: sqr#(0()) -> c_11() 12: sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) 13: terms#(N) -> c_13(sqr#(N)) 14: terms#(X) -> c_14() * Step 3: 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_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_13(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() sqr#(0()) -> c_11() terms#(X) -> c_14() - 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))) 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,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,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_12/3,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,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():12 -->_1 first#(X1,X2) -> c_8():11 2:S:activate#(n__terms(X)) -> c_3(terms#(X)) -->_1 terms#(N) -> c_13(sqr#(N)):7 -->_1 terms#(X) -> c_14():14 3:S:add#(s(X),Y) -> c_5(add#(X,Y)) -->_1 add#(0(),X) -> c_4():9 -->_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():10 -->_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():8 -->_1 activate#(n__terms(X)) -> c_3(terms#(X)):2 -->_1 activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)):1 6:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_2 sqr#(0()) -> c_11():13 -->_3 dbl#(0()) -> c_6():10 -->_1 add#(0(),X) -> c_4():9 -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):6 -->_3 dbl#(s(X)) -> c_7(dbl#(X)):4 -->_1 add#(s(X),Y) -> c_5(add#(X,Y)):3 7:S:terms#(N) -> c_13(sqr#(N)) -->_1 sqr#(0()) -> c_11():13 -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):6 8:W:activate#(X) -> c_1() 9:W:add#(0(),X) -> c_4() 10:W:dbl#(0()) -> c_6() 11:W:first#(X1,X2) -> c_8() 12:W:first#(0(),X) -> c_9() 13:W:sqr#(0()) -> c_11() 14:W:terms#(X) -> c_14() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: first#(X1,X2) -> c_8() 12: first#(0(),X) -> c_9() 14: terms#(X) -> c_14() 9: add#(0(),X) -> c_4() 10: dbl#(0()) -> c_6() 13: sqr#(0()) -> c_11() 8: activate#(X) -> c_1() * Step 4: UsableRules 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_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N)) - 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))) 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,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,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_12/3,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,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#(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_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N)) * Step 5: DecomposeDG 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_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_13(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,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,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_12/3,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, 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_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N)) and a lower component add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(s(X)) -> c_7(dbl#(X)) 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)) -> dbl#(X) sqr#(s(X)) -> sqr#(X) terms#(N) -> sqr#(N) ** Step 5.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict 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_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_13(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,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,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_12/3,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: SimplifyRHS + 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)):3 2:S:activate#(n__terms(X)) -> c_3(terms#(X)) -->_1 terms#(N) -> c_13(sqr#(N)):5 3:S: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:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4 5:S:terms#(N) -> c_13(sqr#(N)) -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sqr#(s(X)) -> c_12(sqr#(X)) ** Step 5.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict 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_12(sqr#(X)) terms#(N) -> c_13(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,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,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_12/1,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: 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_12(sqr#(X)) terms#(N) -> c_13(sqr#(N)) ** Step 5.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict 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_12(sqr#(X)) terms#(N) -> c_13(sqr#(N)) - Signature: {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,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_12/1,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_13) = {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 + [1] p(dbl) = [0] p(first) = [0] p(n__first) = [1] x2 + [0] p(n__terms) = [7] p(nil) = [0] p(recip) = [1] p(s) = [0] p(sqr) = [2] x1 + [2] p(terms) = [8] x1 + [0] p(activate#) = [1] x1 + [0] p(add#) = [1] x2 + [0] p(dbl#) = [8] x1 + [1] p(first#) = [1] x2 + [0] p(sqr#) = [0] p(terms#) = [1] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [0] Following rules are strictly oriented: activate#(n__terms(X)) = [7] > [1] = c_3(terms#(X)) first#(s(X),cons(Y,Z)) = [1] Y + [1] Z + [1] > [1] Z + [0] = c_10(activate#(Z)) terms#(N) = [1] > [0] = c_13(sqr#(N)) Following rules are (at-least) weakly oriented: activate#(n__first(X1,X2)) = [1] X2 + [0] >= [1] X2 + [0] = c_2(first#(X1,X2)) sqr#(s(X)) = [0] >= [0] = c_12(sqr#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 5.a:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) sqr#(s(X)) -> c_12(sqr#(X)) - Weak DPs: activate#(n__terms(X)) -> c_3(terms#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) terms#(N) -> c_13(sqr#(N)) - Signature: {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,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_12/1,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_13) = {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__terms) = [1] x1 + [0] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [1] x1 + [5] p(sqr) = [0] p(terms) = [0] p(activate#) = [5] x1 + [0] p(add#) = [1] p(dbl#) = [0] p(first#) = [5] x2 + [0] p(sqr#) = [5] x1 + [0] p(terms#) = [5] x1 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [0] Following rules are strictly oriented: sqr#(s(X)) = [5] X + [25] > [5] X + [0] = c_12(sqr#(X)) Following rules are (at-least) weakly oriented: activate#(n__first(X1,X2)) = [5] X1 + [5] X2 + [0] >= [5] X2 + [0] = c_2(first#(X1,X2)) activate#(n__terms(X)) = [5] X + [0] >= [5] X + [0] = c_3(terms#(X)) first#(s(X),cons(Y,Z)) = [5] Y + [5] Z + [0] >= [5] Z + [0] = c_10(activate#(Z)) terms#(N) = [5] N + [0] >= [5] N + [0] = c_13(sqr#(N)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 5.a:5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(X1,X2)) - Weak DPs: activate#(n__terms(X)) -> c_3(terms#(X)) first#(s(X),cons(Y,Z)) -> c_10(activate#(Z)) sqr#(s(X)) -> c_12(sqr#(X)) terms#(N) -> c_13(sqr#(N)) - Signature: {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,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_12/1,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_13) = {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 + [1] p(n__terms) = [0] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [1] x1 + [0] p(sqr) = [0] p(terms) = [0] p(activate#) = [9] x1 + [0] p(add#) = [0] p(dbl#) = [8] p(first#) = [9] x1 + [9] x2 + [0] p(sqr#) = [0] p(terms#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [0] Following rules are strictly oriented: activate#(n__first(X1,X2)) = [9] X1 + [9] X2 + [9] > [9] X1 + [9] X2 + [0] = c_2(first#(X1,X2)) Following rules are (at-least) weakly oriented: activate#(n__terms(X)) = [0] >= [0] = c_3(terms#(X)) first#(s(X),cons(Y,Z)) = [9] X + [9] Y + [9] Z + [0] >= [9] Z + [0] = c_10(activate#(Z)) sqr#(s(X)) = [0] >= [0] = c_12(sqr#(X)) terms#(N) = [0] >= [0] = c_13(sqr#(N)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 5.a:6: EmptyProcessor 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_12(sqr#(X)) terms#(N) -> c_13(sqr#(N)) - Signature: {activate/1,add/2,dbl/1,first/2,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,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_12/1,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,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: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_5(add#(X,Y)) dbl#(s(X)) -> c_7(dbl#(X)) - 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)) -> 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,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,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_12/3,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,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 any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_7) = {1} Following symbols are considered usable: {activate#,add#,dbl#,first#,sqr#,terms#} TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [0] p(add) = [5] x1 + [0] p(cons) = [1] x2 + [9] p(dbl) = [4] p(first) = [2] x1 + [1] x2 + [2] p(n__first) = [1] x1 + [1] x2 + [3] p(n__terms) = [1] x1 + [9] p(nil) = [2] p(recip) = [1] x1 + [1] p(s) = [1] x1 + [4] p(sqr) = [3] x1 + [5] p(terms) = [0] p(activate#) = [2] x1 + [3] p(add#) = [0] p(dbl#) = [2] x1 + [0] p(first#) = [1] x1 + [2] x2 + [9] p(sqr#) = [2] x1 + [9] p(terms#) = [2] x1 + [9] p(c_1) = [0] p(c_2) = [8] x1 + [2] p(c_3) = [1] x1 + [1] p(c_4) = [0] p(c_5) = [2] x1 + [0] p(c_6) = [1] p(c_7) = [1] x1 + [4] p(c_8) = [0] p(c_9) = [2] p(c_10) = [2] x1 + [0] p(c_11) = [1] p(c_12) = [1] x2 + [0] p(c_13) = [1] x1 + [1] p(c_14) = [0] Following rules are strictly oriented: dbl#(s(X)) = [2] X + [8] > [2] X + [4] = c_7(dbl#(X)) Following rules are (at-least) weakly oriented: activate#(n__first(X1,X2)) = [2] X1 + [2] X2 + [9] >= [1] X1 + [2] X2 + [9] = first#(X1,X2) activate#(n__terms(X)) = [2] X + [21] >= [2] X + [9] = terms#(X) add#(s(X),Y) = [0] >= [0] = c_5(add#(X,Y)) first#(s(X),cons(Y,Z)) = [1] X + [2] Z + [31] >= [2] Z + [3] = activate#(Z) sqr#(s(X)) = [2] X + [17] >= [0] = add#(sqr(X),dbl(X)) sqr#(s(X)) = [2] X + [17] >= [2] X + [0] = dbl#(X) sqr#(s(X)) = [2] X + [17] >= [2] X + [9] = sqr#(X) terms#(N) = [2] N + [9] >= [2] N + [9] = sqr#(N) ** Step 5.b:2: Ara 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) dbl#(s(X)) -> c_7(dbl#(X)) first#(s(X),cons(Y,Z)) -> activate#(Z) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(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,sqr/1,terms/1,activate#/1,add#/2,dbl#/1,first#/2,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_12/3,c_13/1,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,dbl#,first#,sqr# ,terms#} and constructors {0,cons,n__first,n__terms,nil,recip,s} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(1, 0) 0 :: [] -(0)-> "A"(2, 0) 0 :: [] -(0)-> "A"(7, 3) 0 :: [] -(0)-> "A"(7, 7) add :: ["A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) cons :: ["A"(10, 0) x "A"(10, 10)] -(7)-> "A"(7, 10) dbl :: ["A"(2, 0)] -(4)-> "A"(1, 0) n__first :: ["A"(10, 10) x "A"(10, 10)] -(0)-> "A"(10, 10) n__terms :: ["A"(10, 10)] -(0)-> "A"(10, 10) s :: ["A"(1, 0)] -(1)-> "A"(1, 0) s :: ["A"(2, 0)] -(2)-> "A"(2, 0) s :: ["A"(10, 3)] -(7)-> "A"(7, 3) s :: ["A"(7, 0)] -(7)-> "A"(7, 0) s :: ["A"(4, 3)] -(1)-> "A"(1, 3) s :: ["A"(11, 4)] -(7)-> "A"(7, 4) sqr :: ["A"(7, 3)] -(3)-> "A"(1, 0) activate# :: ["A"(10, 10)] -(13)-> "A"(0, 0) add# :: ["A"(1, 0) x "A"(1, 0)] -(7)-> "A"(12, 1) dbl# :: ["A"(7, 0)] -(9)-> "A"(1, 2) first# :: ["A"(1, 3) x "A"(7, 10)] -(6)-> "A"(0, 0) sqr# :: ["A"(7, 4)] -(12)-> "A"(1, 1) terms# :: ["A"(7, 6)] -(13)-> "A"(1, 1) c_5 :: ["A"(10, 0)] -(0)-> "A"(12, 10) c_7 :: ["A"(0, 0)] -(1)-> "A"(1, 12) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1, 0) "0_A" :: [] -(0)-> "A"(0, 1) "c_5_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_5_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_7_A" :: ["A"(0)] -(1)-> "A"(1, 0) "c_7_A" :: ["A"(0)] -(0)-> "A"(0, 1) "cons_A" :: ["A"(0, 0) x "A"(0, 0)] -(1)-> "A"(1, 0) "cons_A" :: ["A"(1, 0) x "A"(1, 1)] -(0)-> "A"(0, 1) "n__first_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(1, 0) "n__first_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(0, 1) "n__terms_A" :: ["A"(0)] -(0)-> "A"(1, 0) "n__terms_A" :: ["A"(0)] -(0)-> "A"(0, 1) "s_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "s_A" :: ["A"(1, 1)] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: add#(s(X),Y) -> c_5(add#(X,Y)) 2. Weak: WORST_CASE(?,O(n^3))