MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: f(x,Cons(x',xs)) -> f(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),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() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(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() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(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) number42() -> Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Nil())))))))))))))))))))))))))))))))))))))))))) - Weak TRS: f[Ite][False][Ite](False(),x,y) -> Cons(Cons(Nil(),Nil()),y) f[Ite][False][Ite](False(),Cons(x,xs),y) -> xs f[Ite][False][Ite](True(),x,y) -> x f[Ite][False][Ite](True(),x',Cons(x,xs)) -> 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,number42/0} / {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 ,number42} 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#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil())) ,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)) number42#() -> c_8() Weak DPs f[Ite][False][Ite]#(False(),x,y) -> c_9() f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_10() f[Ite][False][Ite]#(True(),x,y) -> c_11() f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_12() g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(g#(x',xs)) and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil())) ,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)) number42#() -> c_8() - Weak DPs: f[Ite][False][Ite]#(False(),x,y) -> c_9() f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_10() f[Ite][False][Ite]#(True(),x,y) -> c_11() f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_12() g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(g#(x',xs)) - Weak TRS: f(x,Cons(x',xs)) -> f(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),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() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Nil())))))))))))))))))))))))))))))))))))))))))) f[Ite][False][Ite](False(),x,y) -> Cons(Cons(Nil(),Nil()),y) f[Ite][False][Ite](False(),Cons(x,xs),y) -> xs f[Ite][False][Ite](True(),x,y) -> x f[Ite][False][Ite](True(),x',Cons(x,xs)) -> 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() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(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) number42() -> Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,Cons(Nil() ,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,number42/0,f#/2,f[Ite][False][Ite]#/3,g#/2 ,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,number42#/0} / {Cons/2,False/0,Nil/0,True/0,c_1/5,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_12/0,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,number42#} and constructors {Cons,False,Nil,True} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: f[Ite][False][Ite](False(),x,y) -> Cons(Cons(Nil(),Nil()),y) f[Ite][False][Ite](False(),Cons(x,xs),y) -> xs f[Ite][False][Ite](True(),x,y) -> x f[Ite][False][Ite](True(),x',Cons(x,xs)) -> xs lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil())) ,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(),x,y) -> c_9() f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_10() f[Ite][False][Ite]#(True(),x,y) -> c_11() f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_12() 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_13(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(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)) number42#() -> c_8() * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil())) ,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)) number42#() -> c_8() - Weak DPs: f[Ite][False][Ite]#(False(),x,y) -> c_9() f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_10() f[Ite][False][Ite]#(True(),x,y) -> c_11() f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_12() g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(g#(x',xs)) - Weak TRS: f[Ite][False][Ite](False(),x,y) -> Cons(Cons(Nil(),Nil()),y) f[Ite][False][Ite](False(),Cons(x,xs),y) -> xs f[Ite][False][Ite](True(),x,y) -> x f[Ite][False][Ite](True(),x',Cons(x,xs)) -> xs lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,number42/0,f#/2,f[Ite][False][Ite]#/3,g#/2 ,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,number42#/0} / {Cons/2,False/0,Nil/0,True/0,c_1/5,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_12/0,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,number42#} and constructors {Cons,False,Nil,True} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,6,8} by application of Pre({2,6,8}) = {1,5,7}. Here rules are labelled as follows: 1: f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil())) ,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: number42#() -> c_8() 9: f[Ite][False][Ite]#(False(),x,y) -> c_9() 10: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_10() 11: f[Ite][False][Ite]#(True(),x,y) -> c_11() 12: f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_12() 13: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(g#(xs,Cons(Cons(Nil(),Nil()),y))) 14: g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(g#(x',xs)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil())) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))) 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#(x,Nil()) -> c_2() f[Ite][False][Ite]#(False(),x,y) -> c_9() f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_10() f[Ite][False][Ite]#(True(),x,y) -> c_11() f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_12() g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(g#(x',xs)) lt0#(x,Nil()) -> c_6() number42#() -> c_8() - Weak TRS: f[Ite][False][Ite](False(),x,y) -> Cons(Cons(Nil(),Nil()),y) f[Ite][False][Ite](False(),Cons(x,xs),y) -> xs f[Ite][False][Ite](True(),x,y) -> x f[Ite][False][Ite](True(),x',Cons(x,xs)) -> xs lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,number42/0,f#/2,f[Ite][False][Ite]#/3,g#/2 ,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,number42#/0} / {Cons/2,False/0,Nil/0,True/0,c_1/5,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_12/0,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,number42#} and constructors {Cons,False,Nil,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil())) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))) -->_5 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):5 -->_3 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):5 -->_4 f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_12():10 -->_2 f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_12():10 -->_4 f[Ite][False][Ite]#(True(),x,y) -> c_11():9 -->_2 f[Ite][False][Ite]#(True(),x,y) -> c_11():9 -->_4 f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_10():8 -->_2 f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_10():8 -->_4 f[Ite][False][Ite]#(False(),x,y) -> c_9():7 -->_2 f[Ite][False][Ite]#(False(),x,y) -> c_9():7 -->_1 f#(x,Nil()) -> c_2():6 -->_1 f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil())) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))):1 2: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_14(g#(x',xs)):12 -->_1 g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(g#(xs,Cons(Cons(Nil(),Nil()),y))):11 -->_2 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):5 3:S:g#(x,Nil()) -> c_4() 4:S:goal#(x,y) -> c_5(f#(x,y),g#(x,y)) -->_1 f#(x,Nil()) -> c_2():6 -->_2 g#(x,Nil()) -> c_4():3 -->_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()))):2 -->_1 f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil())) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))):1 5:S:lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) -->_1 lt0#(x,Nil()) -> c_6():13 -->_1 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):5 6:W:f#(x,Nil()) -> c_2() 7:W:f[Ite][False][Ite]#(False(),x,y) -> c_9() 8:W:f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_10() 9:W:f[Ite][False][Ite]#(True(),x,y) -> c_11() 10:W:f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_12() 11:W:g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(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()))):2 12:W:g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(g#(x',xs)) -->_1 g#(x,Nil()) -> c_4():3 -->_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()))):2 13:W:lt0#(x,Nil()) -> c_6() 14:W:number42#() -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: number42#() -> c_8() 6: f#(x,Nil()) -> c_2() 7: f[Ite][False][Ite]#(False(),x,y) -> c_9() 8: f[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_10() 9: f[Ite][False][Ite]#(True(),x,y) -> c_11() 10: f[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_12() 13: lt0#(x,Nil()) -> c_6() * Step 5: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil())) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))) 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: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(g#(x',xs)) - Weak TRS: f[Ite][False][Ite](False(),x,y) -> Cons(Cons(Nil(),Nil()),y) f[Ite][False][Ite](False(),Cons(x,xs),y) -> xs f[Ite][False][Ite](True(),x,y) -> x f[Ite][False][Ite](True(),x',Cons(x,xs)) -> xs lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,number42/0,f#/2,f[Ite][False][Ite]#/3,g#/2 ,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,number42#/0} / {Cons/2,False/0,Nil/0,True/0,c_1/5,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_12/0,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,number42#} and constructors {Cons,False,Nil,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil())) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))) -->_5 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):5 -->_3 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):5 -->_1 f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil())) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))):1 2: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_14(g#(x',xs)):12 -->_1 g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(g#(xs,Cons(Cons(Nil(),Nil()),y))):11 -->_2 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):5 3:S:g#(x,Nil()) -> c_4() 4:S:goal#(x,y) -> c_5(f#(x,y),g#(x,y)) -->_2 g#(x,Nil()) -> c_4():3 -->_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()))):2 -->_1 f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil())) ,f[Ite][False][Ite]#(lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,lt0#(x,Cons(Nil(),Nil()))):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 11:W:g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(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()))):2 12:W:g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(g#(x',xs)) -->_1 g#(x,Nil()) -> c_4():3 -->_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()))):2 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#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,lt0#(x,Cons(Nil(),Nil())) ,lt0#(x,Cons(Nil(),Nil()))) * Step 6: RemoveHeads MAYBE + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,lt0#(x,Cons(Nil(),Nil())) ,lt0#(x,Cons(Nil(),Nil()))) 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: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(g#(x',xs)) - Weak TRS: f[Ite][False][Ite](False(),x,y) -> Cons(Cons(Nil(),Nil()),y) f[Ite][False][Ite](False(),Cons(x,xs),y) -> xs f[Ite][False][Ite](True(),x,y) -> x f[Ite][False][Ite](True(),x',Cons(x,xs)) -> xs lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,number42/0,f#/2,f[Ite][False][Ite]#/3,g#/2 ,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,number42#/0} / {Cons/2,False/0,Nil/0,True/0,c_1/3,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_12/0,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,number42#} and constructors {Cons,False,Nil,True} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,lt0#(x,Cons(Nil(),Nil())) ,lt0#(x,Cons(Nil(),Nil()))) -->_3 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):5 -->_2 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):5 -->_1 f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,lt0#(x,Cons(Nil(),Nil())) ,lt0#(x,Cons(Nil(),Nil()))):1 2: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_14(g#(x',xs)):7 -->_1 g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(g#(xs,Cons(Cons(Nil(),Nil()),y))):6 -->_2 lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)):5 3:S:g#(x,Nil()) -> c_4() 4:S:goal#(x,y) -> c_5(f#(x,y),g#(x,y)) -->_2 g#(x,Nil()) -> c_4():3 -->_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()))):2 -->_1 f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,lt0#(x,Cons(Nil(),Nil())) ,lt0#(x,Cons(Nil(),Nil()))):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:g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(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()))):2 7:W:g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(g#(x',xs)) -->_1 g#(x,Nil()) -> c_4():3 -->_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()))):2 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: NaturalMI MAYBE + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,lt0#(x,Cons(Nil(),Nil())) ,lt0#(x,Cons(Nil(),Nil()))) 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: g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(g#(x',xs)) - Weak TRS: f[Ite][False][Ite](False(),x,y) -> Cons(Cons(Nil(),Nil()),y) f[Ite][False][Ite](False(),Cons(x,xs),y) -> xs f[Ite][False][Ite](True(),x,y) -> x f[Ite][False][Ite](True(),x',Cons(x,xs)) -> xs lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,number42/0,f#/2,f[Ite][False][Ite]#/3,g#/2 ,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,number42#/0} / {Cons/2,False/0,Nil/0,True/0,c_1/3,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_12/0,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,number42#} and constructors {Cons,False,Nil,True} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_1) = {1,2,3}, uargs(c_3) = {1,2}, uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,number42#} TcT has computed the following interpretation: p(Cons) = [0] p(False) = [0] p(Nil) = [1] p(True) = [0] p(f) = [1] x1 + [4] p(f[Ite][False][Ite]) = [4] p(g) = [0] p(g[Ite][False][Ite]) = [1] x1 + [1] x2 + [0] p(goal) = [2] x1 + [1] x2 + [0] p(lt0) = [0] p(number42) = [1] p(f#) = [0] p(f[Ite][False][Ite]#) = [1] x2 + [2] x3 + [1] p(g#) = [4] p(g[Ite][False][Ite]#) = [2] x3 + [4] p(goal#) = [1] x1 + [4] p(lt0#) = [0] p(number42#) = [0] p(c_1) = [2] x1 + [4] x2 + [4] x3 + [0] p(c_2) = [0] p(c_3) = [1] x1 + [2] x2 + [0] p(c_4) = [3] p(c_5) = [1] x1 + [2] p(c_6) = [4] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] Following rules are strictly oriented: g#(x,Nil()) = [4] > [3] = c_4() Following rules are (at-least) weakly oriented: f#(x,Cons(x',xs)) = [0] >= [0] = c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,lt0#(x,Cons(Nil(),Nil())) ,lt0#(x,Cons(Nil(),Nil()))) g#(x,Cons(x',xs)) = [4] >= [4] = 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) = [2] y + [4] >= [4] = c_13(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) = [4] >= [4] = c_14(g#(x',xs)) lt0#(Cons(x',xs'),Cons(x,xs)) = [0] >= [0] = c_7(lt0#(xs',xs)) * Step 8: Failure MAYBE + Considered Problem: - Strict DPs: f#(x,Cons(x',xs)) -> c_1(f#(f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs)) ,f[Ite][False][Ite](lt0(x,Cons(Nil(),Nil())),x,Cons(x',xs))) ,lt0#(x,Cons(Nil(),Nil())) ,lt0#(x,Cons(Nil(),Nil()))) 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()))) lt0#(Cons(x',xs'),Cons(x,xs)) -> c_7(lt0#(xs',xs)) - Weak DPs: g#(x,Nil()) -> c_4() g[Ite][False][Ite]#(False(),Cons(x,xs),y) -> c_13(g#(xs,Cons(Cons(Nil(),Nil()),y))) g[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(g#(x',xs)) - Weak TRS: f[Ite][False][Ite](False(),x,y) -> Cons(Cons(Nil(),Nil()),y) f[Ite][False][Ite](False(),Cons(x,xs),y) -> xs f[Ite][False][Ite](True(),x,y) -> x f[Ite][False][Ite](True(),x',Cons(x,xs)) -> xs lt0(x,Nil()) -> False() lt0(Cons(x',xs'),Cons(x,xs)) -> lt0(xs',xs) - Signature: {f/2,f[Ite][False][Ite]/3,g/2,g[Ite][False][Ite]/3,goal/2,lt0/2,number42/0,f#/2,f[Ite][False][Ite]#/3,g#/2 ,g[Ite][False][Ite]#/3,goal#/2,lt0#/2,number42#/0} / {Cons/2,False/0,Nil/0,True/0,c_1/3,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_12/0,c_13/1,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0# ,number42#} and constructors {Cons,False,Nil,True} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE