WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(x,Cons(x',xs)) -> f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) f(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) g(x,Cons(x',xs)) -> g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) g(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) goal(x,y) -> Cons(f(x,y),Cons(g(x,y),Nil())) lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() number4(n) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) - Weak TRS: f[Ite][False][Ite](False(),Cons(x,xs),y) -> f(xs,Cons(Cons(Nil(),Nil()),y)) f[Ite][False][Ite](True(),x',Cons(x,xs)) -> f(x',xs) g[Ite][False][Ite](False(),Cons(x,xs),y) -> g(xs,Cons(Cons(Nil(),Nil()),y)) g[Ite][False][Ite](True(),x',Cons(x,xs)) -> g(x',xs) - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1} / {Cons/2,False/0 ,Nil/0,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0 ,notEmpty,number4} and constructors {Cons,False,Nil,True} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g#(x,Nil()) -> c_4() goal#(x,y) -> c_5(f#(x,y),g#(x,y)) lt0#(x,Nil()) -> c_6() lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) lt0#(Nil(),Cons(x',xs)) -> c_8() notEmpty#(Cons(x,xs)) -> c_9() notEmpty#(Nil()) -> c_10() number4#(n) -> c_11() Weak DPs f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g#(x,Nil()) -> c_4() goal#(x,y) -> c_5(f#(x,y),g#(x,y)) lt0#(x,Nil()) -> c_6() lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) lt0#(Nil(),Cons(x',xs)) -> c_8() notEmpty#(Cons(x,xs)) -> c_9() notEmpty#(Nil()) -> c_10() number4#(n) -> c_11() - Strict TRS: f(x,Cons(x',xs)) -> f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) f(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) g(x,Cons(x',xs)) -> g[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) g(x,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) goal(x,y) -> Cons(f(x,y),Cons(g(x,y),Nil())) lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() notEmpty(Cons(x,xs)) -> True() notEmpty(Nil()) -> False() number4(n) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak TRS: f[Ite][False][Ite](False(),Cons(x,xs),y) -> f(xs,Cons(Cons(Nil(),Nil()),y)) f[Ite][False][Ite](True(),x',Cons(x,xs)) -> f(x',xs) g[Ite][False][Ite](False(),Cons(x,xs),y) -> g(xs,Cons(Cons(Nil(),Nil()),y)) g[Ite][False][Ite](True(),x',Cons(x,xs)) -> g(x',xs) - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g#(x,Nil()) -> c_4() g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) goal#(x,y) -> c_5(f#(x,y),g#(x,y)) lt0#(x,Nil()) -> c_6() lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) lt0#(Nil(),Cons(x',xs)) -> c_8() notEmpty#(Cons(x,xs)) -> c_9() notEmpty#(Nil()) -> c_10() number4#(n) -> c_11() * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g#(x,Nil()) -> c_4() goal#(x,y) -> c_5(f#(x,y),g#(x,y)) lt0#(x,Nil()) -> c_6() lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) lt0#(Nil(),Cons(x',xs)) -> c_8() notEmpty#(Cons(x,xs)) -> c_9() notEmpty#(Nil()) -> c_10() number4#(n) -> c_11() - Strict TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs} + 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(f[Ite][False][Ite]#) = {1}, uargs(g[Ite][False][Ite]#) = {1}, uargs(c_1) = {1}, uargs(c_3) = {1}, uargs(c_5) = {1,2}, uargs(c_7) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_15) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(Cons) = [1] x1 + [1] x2 + [1] p(False) = [0] p(Nil) = [3] p(True) = [0] p(f) = [0] p(f[Ite][False][Ite]) = [0] p(g) = [4] x1 + [0] p(g[Ite][False][Ite]) = [1] x1 + [1] x2 + [1] x3 + [1] p(goal) = [2] x1 + [2] p(lt0) = [1] x2 + [0] p(notEmpty) = [0] p(number4) = [0] p(f#) = [1] x1 + [0] p(f[Ite][False][Ite]#) = [1] x1 + [1] x2 + [0] p(g#) = [1] p(g[Ite][False][Ite]#) = [1] x1 + [7] p(goal#) = [3] x1 + [0] p(lt0#) = [2] x1 + [5] p(notEmpty#) = [1] p(number4#) = [0] p(c_1) = [1] x1 + [3] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [0] p(c_7) = [1] x1 + [3] p(c_8) = [0] p(c_9) = [2] p(c_10) = [0] p(c_11) = [1] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [5] p(c_15) = [1] x1 + [6] Following rules are strictly oriented: g#(x,Nil()) = [1] > [0] = c_4() lt0#(x,Nil()) = [2] x + [5] > [0] = c_6() lt0#(Nil(),Cons(x',xs)) = [11] > [0] = c_8() notEmpty#(Nil()) = [1] > [0] = c_10() lt0(x,Nil()) = [3] > [0] = False() lt0(Cons(x',xs'),Cons(x,xs)) = [1] x + [1] xs + [1] > [1] xs + [0] = lt0(xs',xs) lt0(Nil(),Cons(x',xs)) = [1] x' + [1] xs + [1] > [0] = True() Following rules are (at-least) weakly oriented: f#(x,Cons(x',xs)) = [1] x + [0] >= [1] x + [10] = c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) = [1] x + [0] >= [0] = c_2() f[Ite][False][Ite]#(False(),Cons(x,xs),y) = [1] x + [1] xs + [1] >= [1] xs + [0] = c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [1] x' + [0] >= [1] x' + [0] = c_13(f#(x',xs)) g#(x,Cons(x',xs)) = [1] >= [14] = c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g[Ite][False][Ite]#(False(),Cons(x,xs),y) = [7] >= [6] = c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [7] >= [7] = c_15(g#(x',xs)) goal#(x,y) = [3] x + [0] >= [1] x + [1] = c_5(f#(x,y),g#(x,y)) lt0#(Cons(x',xs'),Cons(x,xs)) = [2] x' + [2] xs' + [7] >= [2] xs' + [8] = c_7(lt0#(xs',xs)) notEmpty#(Cons(x,xs)) = [1] >= [2] = c_9() number4#(n) = [0] >= [1] = c_11() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) goal#(x,y) -> c_5(f#(x,y),g#(x,y)) lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) notEmpty#(Cons(x,xs)) -> c_9() number4#(n) -> c_11() - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) g#(x,Nil()) -> c_4() g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) lt0#(x,Nil()) -> c_6() lt0#(Nil(),Cons(x',xs)) -> c_8() notEmpty#(Nil()) -> c_10() - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {6,7} by application of Pre({6,7}) = {}. Here rules are labelled as follows: 1: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) 2: f#(x,Nil()) -> c_2() 3: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) 4: goal#(x,y) -> c_5(f#(x,y),g#(x,y)) 5: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) 6: notEmpty#(Cons(x,xs)) -> c_9() 7: number4#(n) -> c_11() 8: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) 9: f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) 10: g#(x,Nil()) -> c_4() 11: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) 12: g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) 13: lt0#(x,Nil()) -> c_6() 14: lt0#(Nil(),Cons(x',xs)) -> c_8() 15: notEmpty#(Nil()) -> c_10() * Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) goal#(x,y) -> c_5(f#(x,y),g#(x,y)) lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) g#(x,Nil()) -> c_4() g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) lt0#(x,Nil()) -> c_6() lt0#(Nil(),Cons(x',xs)) -> c_8() notEmpty#(Cons(x,xs)) -> c_9() notEmpty#(Nil()) -> c_10() number4#(n) -> c_11() - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) -->_1 f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)):7 -->_1 f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))):6 2:S:f#(x,Nil()) -> c_2() 3:S:g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) -->_1 g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)):10 -->_1 g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))):9 4:S:goal#(x,y) -> c_5(f#(x,y),g#(x,y)) -->_2 g#(x,Nil()) -> c_4():8 -->_2 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):3 -->_1 f#(x,Nil()) -> c_2():2 -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 5:S:lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) -->_1 lt0#(Nil(),Cons(x',xs)) -> c_8():12 -->_1 lt0#(x,Nil()) -> c_6():11 -->_1 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):5 6:W:f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 7:W:f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) -->_1 f#(x,Nil()) -> c_2():2 -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 8:W:g#(x,Nil()) -> c_4() 9:W:g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):3 10:W:g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) -->_1 g#(x,Nil()) -> c_4():8 -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):3 11:W:lt0#(x,Nil()) -> c_6() 12:W:lt0#(Nil(),Cons(x',xs)) -> c_8() 13:W:notEmpty#(Cons(x,xs)) -> c_9() 14:W:notEmpty#(Nil()) -> c_10() 15:W:number4#(n) -> c_11() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 15: number4#(n) -> c_11() 14: notEmpty#(Nil()) -> c_10() 13: notEmpty#(Cons(x,xs)) -> c_9() 11: lt0#(x,Nil()) -> c_6() 12: lt0#(Nil(),Cons(x',xs)) -> c_8() 8: g#(x,Nil()) -> c_4() * Step 6: RemoveHeads WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) goal#(x,y) -> c_5(f#(x,y),g#(x,y)) lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) -->_1 f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)):7 -->_1 f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))):6 2:S:f#(x,Nil()) -> c_2() 3:S:g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) -->_1 g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)):10 -->_1 g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))):9 4:S:goal#(x,y) -> c_5(f#(x,y),g#(x,y)) -->_2 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):3 -->_1 f#(x,Nil()) -> c_2():2 -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 5:S:lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) -->_1 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):5 6:W:f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 7:W:f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) -->_1 f#(x,Nil()) -> c_2():2 -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 9:W:g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):3 10:W:g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):3 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(4,goal#(x,y) -> c_5(f#(x,y),g#(x,y)))] * Step 7: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + 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: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal# ,lt0#,notEmpty#,number4#} and constructors {Cons,False,Nil,True} Problem (S) - Strict DPs: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal# ,lt0#,notEmpty#,number4#} and constructors {Cons,False,Nil,True} ** Step 7.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) -->_1 f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)):7 -->_1 f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))):6 2:S:f#(x,Nil()) -> c_2() 3:W:g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) -->_1 g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)):10 -->_1 g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))):9 5:W:lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) -->_1 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):5 6:W:f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 7:W:f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 -->_1 f#(x,Nil()) -> c_2():2 9:W:g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):3 10:W:g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) 3: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) 10: g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) 9: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) ** Step 7.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + 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: 2: f#(x,Nil()) -> c_2() The strictly oriented rules are moved into the weak component. *** Step 7.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + 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_1) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1} Following symbols are considered usable: {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty#,number4#} TcT has computed the following interpretation: p(Cons) = [1] x1 + [1] p(False) = [8] p(Nil) = [4] p(True) = [0] p(f) = [1] x1 + [1] p(f[Ite][False][Ite]) = [2] x1 + [1] x2 + [1] p(g) = [1] x2 + [1] p(g[Ite][False][Ite]) = [2] x1 + [1] x2 + [1] p(goal) = [1] x1 + [1] x2 + [0] p(lt0) = [2] x1 + [4] x2 + [2] p(notEmpty) = [1] x1 + [0] p(number4) = [1] x1 + [1] p(f#) = [14] p(f[Ite][False][Ite]#) = [14] p(g#) = [8] x1 + [1] x2 + [0] p(g[Ite][False][Ite]#) = [1] x3 + [1] p(goal#) = [1] x1 + [1] p(lt0#) = [2] x2 + [1] p(notEmpty#) = [0] p(number4#) = [1] x1 + [2] p(c_1) = [1] x1 + [0] p(c_2) = [9] p(c_3) = [0] p(c_4) = [0] p(c_5) = [2] x1 + [2] x2 + [4] p(c_6) = [1] p(c_7) = [0] p(c_8) = [1] p(c_9) = [1] p(c_10) = [0] p(c_11) = [1] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] x1 + [0] Following rules are strictly oriented: f#(x,Nil()) = [14] > [9] = c_2() Following rules are (at-least) weakly oriented: f#(x,Cons(x',xs)) = [14] >= [14] = c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f[Ite][False][Ite]#(False(),Cons(x,xs),y) = [14] >= [14] = c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [14] >= [14] = c_13(f#(x',xs)) *** Step 7.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) - Weak DPs: f#(x,Nil()) -> c_2() f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 7.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) - Weak DPs: f#(x,Nil()) -> c_2() f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) -->_1 f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)):4 -->_1 f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))):3 2:W:f#(x,Nil()) -> c_2() 3:W:f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 4:W:f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) -->_1 f#(x,Nil()) -> c_2():2 -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: f#(x,Nil()) -> c_2() *** Step 7.a:2.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + 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: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) Consider the set of all dependency pairs 1: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) 3: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) 4: f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,3,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 7.a:2.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + 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_1) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1} Following symbols are considered usable: {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty#,number4#} TcT has computed the following interpretation: p(Cons) = [1] x2 + [1] p(False) = [4] p(Nil) = [0] p(True) = [0] p(f) = [1] x1 + [1] p(f[Ite][False][Ite]) = [1] x1 + [0] p(g) = [1] x1 + [1] p(g[Ite][False][Ite]) = [2] x2 + [4] p(goal) = [1] x1 + [1] x2 + [1] p(lt0) = [12] p(notEmpty) = [1] x1 + [1] p(number4) = [1] x1 + [8] p(f#) = [6] x1 + [2] x2 + [7] p(f[Ite][False][Ite]#) = [6] x2 + [2] x3 + [6] p(g#) = [1] x1 + [0] p(g[Ite][False][Ite]#) = [1] x1 + [1] x2 + [0] p(goal#) = [1] p(lt0#) = [4] x1 + [1] x2 + [1] p(notEmpty#) = [1] x1 + [1] p(number4#) = [0] p(c_1) = [1] x1 + [0] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [4] p(c_5) = [8] x1 + [1] x2 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [2] p(c_9) = [1] p(c_10) = [0] p(c_11) = [1] p(c_12) = [1] x1 + [3] p(c_13) = [1] x1 + [1] p(c_14) = [1] p(c_15) = [1] x1 + [0] Following rules are strictly oriented: f#(x,Cons(x',xs)) = [6] x + [2] xs + [9] > [6] x + [2] xs + [8] = c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) Following rules are (at-least) weakly oriented: f[Ite][False][Ite]#(False(),Cons(x,xs),y) = [6] xs + [2] y + [12] >= [6] xs + [2] y + [12] = c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [6] x' + [2] xs + [8] >= [6] x' + [2] xs + [8] = c_13(f#(x',xs)) **** Step 7.a:2.b:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 7.a:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) -->_1 f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)):3 -->_1 f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))):2 2:W:f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 3:W:f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) 3: f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) 2: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) **** Step 7.a:2.b:2.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f#(x,Nil()) -> c_2() f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) -->_1 g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)):8 -->_1 g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))):7 2:S:lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) -->_1 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):2 3:W:f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) -->_1 f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)):6 -->_1 f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))):5 4:W:f#(x,Nil()) -> c_2() 5:W:f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):3 6:W:f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) -->_1 f#(x,Nil()) -> c_2():4 -->_1 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):3 7:W:g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 8:W:g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) 6: f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) 5: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) 4: f#(x,Nil()) -> c_2() ** Step 7.b:2: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak DPs: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + 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: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) - Weak DPs: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal# ,lt0#,notEmpty#,number4#} and constructors {Cons,False,Nil,True} Problem (S) - Strict DPs: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak DPs: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal# ,lt0#,notEmpty#,number4#} and constructors {Cons,False,Nil,True} *** Step 7.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) - Weak DPs: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) -->_1 g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)):8 -->_1 g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))):7 2:W:lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) -->_1 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):2 7:W:g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 8:W:g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) *** Step 7.b:2.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) - Weak DPs: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + 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: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) Consider the set of all dependency pairs 1: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) 7: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) 8: g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,7,8} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 7.b:2.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) - Weak DPs: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + 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_3) = {1}, uargs(c_14) = {1}, uargs(c_15) = {1} Following symbols are considered usable: {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty#,number4#} TcT has computed the following interpretation: p(Cons) = [1] x2 + [2] p(False) = [0] p(Nil) = [1] p(True) = [0] p(f) = [1] x2 + [2] p(f[Ite][False][Ite]) = [1] x1 + [2] x3 + [0] p(g) = [1] x2 + [2] p(g[Ite][False][Ite]) = [8] x1 + [2] x2 + [1] x3 + [2] p(goal) = [1] x1 + [0] p(lt0) = [8] x2 + [0] p(notEmpty) = [1] x1 + [1] p(number4) = [2] p(f#) = [1] x2 + [1] p(f[Ite][False][Ite]#) = [1] x1 + [1] p(g#) = [15] x1 + [8] x2 + [2] p(g[Ite][False][Ite]#) = [15] x2 + [8] x3 + [0] p(goal#) = [2] x1 + [2] p(lt0#) = [1] x2 + [0] p(notEmpty#) = [1] p(number4#) = [1] x1 + [1] p(c_1) = [1] x1 + [8] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [1] x2 + [0] p(c_6) = [0] p(c_7) = [1] x1 + [2] p(c_8) = [0] p(c_9) = [1] p(c_10) = [0] p(c_11) = [2] p(c_12) = [1] x1 + [4] p(c_13) = [0] p(c_14) = [1] x1 + [12] p(c_15) = [1] x1 + [14] Following rules are strictly oriented: g#(x,Cons(x',xs)) = [15] x + [8] xs + [18] > [15] x + [8] xs + [16] = c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) Following rules are (at-least) weakly oriented: g[Ite][False][Ite]#(False(),Cons(x,xs),y) = [15] xs + [8] y + [30] >= [15] xs + [8] y + [30] = c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [15] x' + [8] xs + [16] >= [15] x' + [8] xs + [16] = c_15(g#(x',xs)) **** Step 7.b:2.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 7.b:2.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) -->_1 g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)):3 -->_1 g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))):2 2:W:g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 3:W:g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) 3: g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) 2: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) **** Step 7.b:2.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 7.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak DPs: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) -->_1 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):1 2:W:g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) -->_1 g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)):4 -->_1 g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))):3 3:W:g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):2 4:W:g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) 4: g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) 3: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) *** Step 7.b:2.b:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak TRS: lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) lt0(Nil(),Cons(x',xs)) -> True() - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) *** Step 7.b:2.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + 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: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) The strictly oriented rules are moved into the weak component. **** Step 7.b:2.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + 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_7) = {1} Following symbols are considered usable: {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty#,number4#} TcT has computed the following interpretation: p(Cons) = [1] x1 + [1] x2 + [8] p(False) = [0] p(Nil) = [0] p(True) = [0] p(f) = [0] p(f[Ite][False][Ite]) = [0] p(g) = [0] p(g[Ite][False][Ite]) = [0] p(goal) = [0] p(lt0) = [0] p(notEmpty) = [0] p(number4) = [0] p(f#) = [0] p(f[Ite][False][Ite]#) = [1] x2 + [1] x3 + [0] p(g#) = [1] x1 + [1] p(g[Ite][False][Ite]#) = [1] x2 + [0] p(goal#) = [1] x1 + [0] p(lt0#) = [2] x1 + [14] p(notEmpty#) = [1] p(number4#) = [2] x1 + [8] p(c_1) = [1] x1 + [0] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [1] x2 + [0] p(c_6) = [0] p(c_7) = [1] x1 + [14] p(c_8) = [4] p(c_9) = [0] p(c_10) = [1] p(c_11) = [1] p(c_12) = [1] x1 + [1] p(c_13) = [1] p(c_14) = [0] p(c_15) = [2] x1 + [0] Following rules are strictly oriented: lt0#(Cons(x',xs'),Cons(x,xs)) = [2] x' + [2] xs' + [30] > [2] xs' + [28] = c_7(lt0#(xs',xs)) Following rules are (at-least) weakly oriented: **** Step 7.b:2.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 7.b:2.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) -->_1 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) **** Step 7.b:2.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,notEmpty/1,number4/1,f#/2 ,f[Ite][False][Ite]#/3,g#/2,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,notEmpty#/1,number4#/1} / {Cons/2,False/0 ,Nil/0,True/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1 ,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,notEmpty#,number4#} and constructors {Cons,False,Nil,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))