WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + 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_ = DT} + Details: We add the following dependency tuples: Strict DPs f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))) 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#(x,Cons(Nil(),Nil()))) 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: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))) 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#(x,Cons(Nil(),Nil()))) 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)) - Weak 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())))) 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(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())))) 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) 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())))) - 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_1/2,c_2/0,c_3/2,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 {6,8,9,10,11} by application of Pre({6,8,9,10,11}) = {1,3,7}. 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)) ,lt0#(x,Cons(Nil(),Nil()))) 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)) ,lt0#(x,Cons(Nil(),Nil()))) 4: g#(x,Nil()) -> c_4() 5: goal#(x,y) -> c_5(f#(x,y),g#(x,y)) 6: lt0#(x,Nil()) -> c_6() 7: lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) 8: lt0#(Nil(),Cons(x',xs)) -> c_8() 9: notEmpty#(Cons(x,xs)) -> c_9() 10: notEmpty#(Nil()) -> c_10() 11: number4#(n) -> c_11() 12: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) 13: f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)) 14: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) 15: g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))) 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#(x,Cons(Nil(),Nil()))) g#(x,Nil()) -> c_4() 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)) 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: 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())))) 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(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())))) 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) 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())))) - 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_1/2,c_2/0,c_3/2,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)) ,lt0#(x,Cons(Nil(),Nil()))) -->_1 f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)):8 -->_1 f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))):7 -->_2 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):6 -->_2 lt0#(Nil(),Cons(x',xs)) -> c_8():12 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)) ,lt0#(x,Cons(Nil(),Nil()))) -->_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 -->_2 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):6 -->_2 lt0#(Nil(),Cons(x',xs)) -> c_8():12 4:S:g#(x,Nil()) -> c_4() 5:S:goal#(x,y) -> c_5(f#(x,y),g#(x,y)) -->_2 g#(x,Nil()) -> c_4():4 -->_2 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))):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)) ,lt0#(x,Cons(Nil(),Nil()))):1 6: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)):6 7: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)) ,lt0#(x,Cons(Nil(),Nil()))):1 8: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)) ,lt0#(x,Cons(Nil(),Nil()))):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)) ,lt0#(x,Cons(Nil(),Nil()))):3 10:W:g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) -->_1 g#(x,Nil()) -> c_4():4 -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))):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() * Step 4: RemoveHeads WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))) 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#(x,Cons(Nil(),Nil()))) g#(x,Nil()) -> c_4() 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: 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())))) 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(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())))) 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) 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())))) - 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_1/2,c_2/0,c_3/2,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)) ,lt0#(x,Cons(Nil(),Nil()))) -->_1 f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)):8 -->_1 f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))):7 -->_2 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):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)) ,lt0#(x,Cons(Nil(),Nil()))) -->_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 -->_2 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):6 4:S:g#(x,Nil()) -> c_4() 5:S:goal#(x,y) -> c_5(f#(x,y),g#(x,y)) -->_2 g#(x,Nil()) -> c_4():4 -->_2 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))):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)) ,lt0#(x,Cons(Nil(),Nil()))):1 6: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)):6 7: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)) ,lt0#(x,Cons(Nil(),Nil()))):1 8: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)) ,lt0#(x,Cons(Nil(),Nil()))):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)) ,lt0#(x,Cons(Nil(),Nil()))):3 10:W:g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) -->_1 g#(x,Nil()) -> c_4():4 -->_1 g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))):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). [(5,goal#(x,y) -> c_5(f#(x,y),g#(x,y)))] * Step 5: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))) 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#(x,Cons(Nil(),Nil()))) g#(x,Nil()) -> c_4() 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: 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())))) 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(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())))) 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) 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())))) - 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_1/2,c_2/0,c_3/2,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)) ,lt0#(x,Cons(Nil(),Nil()))) 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)) ,lt0#(x,Cons(Nil(),Nil()))) 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#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) * Step 6: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))) 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#(x,Cons(Nil(),Nil()))) g#(x,Nil()) -> c_4() 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_1/2,c_2/0,c_3/2,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: 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 f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))) 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)) ,lt0#(x,Cons(Nil(),Nil()))) 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 a lower component f#(x,Nil()) -> c_2() g#(x,Nil()) -> c_4() lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) Further, following extension rules are added to the lower component. f#(x,Cons(x',xs)) -> f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) f#(x,Cons(x',xs)) -> lt0#(x,Cons(Nil(),Nil())) 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#(x,Cons(x',xs)) -> g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) g#(x,Cons(x',xs)) -> lt0#(x,Cons(Nil(),Nil())) 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) ** Step 6.a:1: SimplifyRHS 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)) ,lt0#(x,Cons(Nil(),Nil()))) 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)) ,lt0#(x,Cons(Nil(),Nil()))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) - 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_1/2,c_2/0,c_3/2,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: SimplifyRHS + 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)) ,lt0#(x,Cons(Nil(),Nil()))) -->_1 f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))):5 -->_1 f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_13(f#(x',xs)):2 2:S: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)) ,lt0#(x,Cons(Nil(),Nil()))):1 3:S:g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))) -->_1 g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))):6 -->_1 g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)):4 4:S: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)) ,lt0#(x,Cons(Nil(),Nil()))):3 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)) ,lt0#(x,Cons(Nil(),Nil()))):1 6: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)) ,lt0#(x,Cons(Nil(),Nil()))):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: f#(x,Cons(x',xs)) -> c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g#(x,Cons(x',xs)) -> c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ** Step 6.a:2: 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[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]#(True(),x',Cons(x,xs)) -> c_15(g#(x',xs)) - Weak DPs: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) - 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: 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(f[Ite][False][Ite]#) = {1}, uargs(g[Ite][False][Ite]#) = {1}, uargs(c_1) = {1}, uargs(c_3) = {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] x2 + [4] p(False) = [1] p(Nil) = [5] p(True) = [1] p(f) = [0] p(f[Ite][False][Ite]) = [0] p(g) = [0] p(g[Ite][False][Ite]) = [0] p(goal) = [0] p(lt0) = [1] x2 + [0] p(notEmpty) = [0] p(number4) = [0] p(f#) = [0] p(f[Ite][False][Ite]#) = [1] x1 + [1] p(g#) = [1] p(g[Ite][False][Ite]#) = [1] x1 + [2] p(goal#) = [0] p(lt0#) = [0] p(notEmpty#) = [0] p(number4#) = [1] x1 + [2] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [1] x1 + [1] p(c_4) = [2] p(c_5) = [1] x1 + [1] x2 + [2] p(c_6) = [0] p(c_7) = [1] x1 + [1] p(c_8) = [4] p(c_9) = [1] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [1] p(c_15) = [1] x1 + [0] Following rules are strictly oriented: f[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [2] > [0] = c_13(f#(x',xs)) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [3] > [1] = c_15(g#(x',xs)) Following rules are (at-least) weakly oriented: f#(x,Cons(x',xs)) = [0] >= [10] = c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f[Ite][False][Ite]#(False(),Cons(x,xs),y) = [2] >= [0] = c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) g#(x,Cons(x',xs)) = [1] >= [12] = c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g[Ite][False][Ite]#(False(),Cons(x,xs),y) = [3] >= [2] = c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) lt0(x,Nil()) = [5] >= [1] = False() lt0(Cons(x',xs'),Cons(x,xs)) = [1] xs + [4] >= [1] xs + [0] = lt0(xs',xs) lt0(Nil(),Cons(x',xs)) = [1] xs + [4] >= [1] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a: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))) g#(x,Cons(x',xs)) -> c_3(g[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)) 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: 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(f[Ite][False][Ite]#) = {1}, uargs(g[Ite][False][Ite]#) = {1}, uargs(c_1) = {1}, uargs(c_3) = {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] x2 + [4] p(False) = [0] p(Nil) = [4] p(True) = [0] p(f) = [1] x1 + [1] p(f[Ite][False][Ite]) = [4] x2 + [1] p(g) = [2] x1 + [1] p(g[Ite][False][Ite]) = [1] x1 + [2] x2 + [1] x3 + [0] p(goal) = [1] x1 + [1] x2 + [0] p(lt0) = [0] p(notEmpty) = [1] x1 + [1] p(number4) = [4] x1 + [2] p(f#) = [1] p(f[Ite][False][Ite]#) = [1] x1 + [4] p(g#) = [3] x1 + [2] x2 + [2] p(g[Ite][False][Ite]#) = [1] x1 + [3] x2 + [2] x3 + [0] p(goal#) = [2] x1 + [1] x2 + [0] p(lt0#) = [4] x2 + [1] p(notEmpty#) = [1] p(number4#) = [4] x1 + [0] p(c_1) = [1] x1 + [0] p(c_2) = [1] p(c_3) = [1] x1 + [1] p(c_4) = [2] p(c_5) = [4] x2 + [0] p(c_6) = [1] p(c_7) = [1] p(c_8) = [0] p(c_9) = [1] p(c_10) = [4] p(c_11) = [1] p(c_12) = [1] x1 + [3] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] x1 + [5] Following rules are strictly oriented: g#(x,Cons(x',xs)) = [3] x + [2] xs + [10] > [3] x + [2] xs + [9] = c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) Following rules are (at-least) weakly oriented: f#(x,Cons(x',xs)) = [1] >= [4] = c_1(f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) f[Ite][False][Ite]#(False(),Cons(x,xs),y) = [4] >= [4] = c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [4] >= [1] = c_13(f#(x',xs)) g[Ite][False][Ite]#(False(),Cons(x,xs),y) = [3] xs + [2] y + [12] >= [3] xs + [2] y + [10] = c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [3] x' + [2] xs + [8] >= [3] x' + [2] xs + [7] = c_15(g#(x',xs)) lt0(x,Nil()) = [0] >= [0] = False() lt0(Cons(x',xs'),Cons(x,xs)) = [0] >= [0] = lt0(xs',xs) lt0(Nil(),Cons(x',xs)) = [0] >= [0] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:4: 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))) - 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)) - 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: 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(f[Ite][False][Ite]#) = {1}, uargs(g[Ite][False][Ite]#) = {1}, uargs(c_1) = {1}, uargs(c_3) = {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] x2 + [2] p(False) = [0] p(Nil) = [4] p(True) = [0] p(f) = [1] x1 + [0] p(f[Ite][False][Ite]) = [1] x1 + [1] x2 + [4] x3 + [1] p(g) = [1] x1 + [4] p(g[Ite][False][Ite]) = [1] x1 + [1] p(goal) = [1] x1 + [4] x2 + [4] p(lt0) = [0] p(notEmpty) = [2] x1 + [1] p(number4) = [1] x1 + [2] p(f#) = [2] x1 + [1] x2 + [1] p(f[Ite][False][Ite]#) = [1] x1 + [2] x2 + [1] x3 + [0] p(g#) = [0] p(g[Ite][False][Ite]#) = [1] x1 + [0] p(goal#) = [2] x1 + [2] p(lt0#) = [2] x1 + [1] x2 + [4] p(notEmpty#) = [2] p(number4#) = [0] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [1] x1 + [1] x2 + [1] p(c_6) = [1] p(c_7) = [4] x1 + [4] p(c_8) = [0] p(c_9) = [4] p(c_10) = [0] p(c_11) = [1] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [1] p(c_14) = [1] x1 + [0] p(c_15) = [1] x1 + [0] Following rules are strictly oriented: f#(x,Cons(x',xs)) = [2] x + [1] xs + [3] > [2] x + [1] xs + [2] = 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) = [2] xs + [1] y + [4] >= [2] xs + [1] y + [3] = c_12(f#(xs,Cons(Cons(Nil(),Nil()),y))) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [2] x' + [1] xs + [2] >= [2] x' + [1] xs + [2] = c_13(f#(x',xs)) g#(x,Cons(x',xs)) = [0] >= [0] = c_3(g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) g[Ite][False][Ite]#(False(),Cons(x,xs),y) = [0] >= [0] = c_14(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [0] >= [0] = c_15(g#(x',xs)) lt0(x,Nil()) = [0] >= [0] = False() lt0(Cons(x',xs'),Cons(x,xs)) = [0] >= [0] = lt0(xs',xs) lt0(Nil(),Cons(x',xs)) = [0] >= [0] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:5: EmptyProcessor 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)) 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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Nil()) -> c_2() g#(x,Nil()) -> c_4() lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak DPs: f#(x,Cons(x',xs)) -> f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) f#(x,Cons(x',xs)) -> lt0#(x,Cons(Nil(),Nil())) 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#(x,Cons(x',xs)) -> g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) g#(x,Cons(x',xs)) -> lt0#(x,Cons(Nil(),Nil())) 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) - 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_1/2,c_2/0,c_3/2,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 = 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(f[Ite][False][Ite]#) = {1}, uargs(g[Ite][False][Ite]#) = {1}, uargs(c_7) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(Cons) = [0] p(False) = [0] p(Nil) = [0] p(True) = [0] p(f) = [1] x1 + [1] x2 + [1] p(f[Ite][False][Ite]) = [1] x2 + [4] p(g) = [1] x1 + [2] p(g[Ite][False][Ite]) = [2] x2 + [1] x3 + [1] p(goal) = [1] x2 + [2] p(lt0) = [0] p(notEmpty) = [2] p(number4) = [2] p(f#) = [0] p(f[Ite][False][Ite]#) = [1] x1 + [0] p(g#) = [2] p(g[Ite][False][Ite]#) = [1] x1 + [1] x3 + [2] p(goal#) = [4] x1 + [0] p(lt0#) = [0] p(notEmpty#) = [0] p(number4#) = [0] p(c_1) = [4] x1 + [4] p(c_2) = [2] p(c_3) = [1] x2 + [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [2] p(c_7) = [1] x1 + [4] p(c_8) = [4] p(c_9) = [2] p(c_10) = [0] p(c_11) = [1] p(c_12) = [1] x1 + [1] p(c_13) = [4] p(c_14) = [1] x1 + [1] p(c_15) = [4] x1 + [2] Following rules are strictly oriented: g#(x,Nil()) = [2] > [0] = c_4() Following rules are (at-least) weakly oriented: f#(x,Cons(x',xs)) = [0] >= [0] = f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) f#(x,Cons(x',xs)) = [0] >= [0] = lt0#(x,Cons(Nil(),Nil())) f#(x,Nil()) = [0] >= [2] = c_2() f[Ite][False][Ite]#(False(),Cons(x,xs),y) = [0] >= [0] = f#(xs,Cons(Cons(Nil(),Nil()),y)) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [0] >= [0] = f#(x',xs) g#(x,Cons(x',xs)) = [2] >= [2] = g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) g#(x,Cons(x',xs)) = [2] >= [0] = lt0#(x,Cons(Nil(),Nil())) g[Ite][False][Ite]#(False(),Cons(x,xs),y) = [1] y + [2] >= [2] = g#(xs,Cons(Cons(Nil(),Nil()),y)) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [2] >= [2] = g#(x',xs) lt0#(Cons(x',xs'),Cons(x,xs)) = [0] >= [4] = c_7(lt0#(xs',xs)) lt0(x,Nil()) = [0] >= [0] = False() lt0(Cons(x',xs'),Cons(x,xs)) = [0] >= [0] = lt0(xs',xs) lt0(Nil(),Cons(x',xs)) = [0] >= [0] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.b:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(x,Nil()) -> c_2() lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak DPs: f#(x,Cons(x',xs)) -> f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) f#(x,Cons(x',xs)) -> lt0#(x,Cons(Nil(),Nil())) 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#(x,Cons(x',xs)) -> g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) g#(x,Cons(x',xs)) -> lt0#(x,Cons(Nil(),Nil())) g#(x,Nil()) -> c_4() 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) - 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_1/2,c_2/0,c_3/2,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 = 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(f[Ite][False][Ite]#) = {1}, uargs(g[Ite][False][Ite]#) = {1}, uargs(c_7) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(Cons) = [1] x2 + [4] p(False) = [4] p(Nil) = [0] p(True) = [4] p(f) = [0] p(f[Ite][False][Ite]) = [1] x1 + [1] p(g) = [4] x1 + [4] p(g[Ite][False][Ite]) = [1] x2 + [0] p(goal) = [1] x2 + [4] p(lt0) = [4] p(notEmpty) = [2] p(number4) = [2] x1 + [0] p(f#) = [6] p(f[Ite][False][Ite]#) = [1] x1 + [2] p(g#) = [2] x1 + [1] x2 + [7] p(g[Ite][False][Ite]#) = [1] x1 + [2] x2 + [1] x3 + [2] p(goal#) = [1] x1 + [1] x2 + [0] p(lt0#) = [1] x2 + [2] p(notEmpty#) = [1] p(number4#) = [2] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [7] p(c_5) = [1] x1 + [4] x2 + [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [4] p(c_10) = [0] p(c_11) = [2] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] Following rules are strictly oriented: f#(x,Nil()) = [6] > [0] = c_2() lt0#(Cons(x',xs'),Cons(x,xs)) = [1] xs + [6] > [1] xs + [2] = c_7(lt0#(xs',xs)) Following rules are (at-least) weakly oriented: f#(x,Cons(x',xs)) = [6] >= [6] = f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) f#(x,Cons(x',xs)) = [6] >= [6] = lt0#(x,Cons(Nil(),Nil())) f[Ite][False][Ite]#(False(),Cons(x,xs),y) = [6] >= [6] = f#(xs,Cons(Cons(Nil(),Nil()),y)) f[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [6] >= [6] = f#(x',xs) g#(x,Cons(x',xs)) = [2] x + [1] xs + [11] >= [2] x + [1] xs + [10] = g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) g#(x,Cons(x',xs)) = [2] x + [1] xs + [11] >= [6] = lt0#(x,Cons(Nil(),Nil())) g#(x,Nil()) = [2] x + [7] >= [7] = c_4() g[Ite][False][Ite]#(False(),Cons(x,xs),y) = [2] xs + [1] y + [14] >= [2] xs + [1] y + [11] = g#(xs,Cons(Cons(Nil(),Nil()),y)) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [2] x' + [1] xs + [10] >= [2] x' + [1] xs + [7] = g#(x',xs) lt0(x,Nil()) = [4] >= [4] = False() lt0(Cons(x',xs'),Cons(x,xs)) = [4] >= [4] = lt0(xs',xs) lt0(Nil(),Cons(x',xs)) = [4] >= [4] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f#(x,Cons(x',xs)) -> f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) f#(x,Cons(x',xs)) -> lt0#(x,Cons(Nil(),Nil())) f#(x,Nil()) -> c_2() 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#(x,Cons(x',xs)) -> g[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) g#(x,Cons(x',xs)) -> lt0#(x,Cons(Nil(),Nil())) g#(x,Nil()) -> c_4() 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) 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_1/2,c_2/0,c_3/2,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^2))