WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: F(z0,Cons(z1,z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) F(z0,Nil()) -> c9() G(z0,Cons(z1,z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) G(z0,Nil()) -> c7() GOAL(z0,z1) -> c14(F(z0,z1)) GOAL(z0,z1) -> c15(G(z0,z1)) LT0(z0,Nil()) -> c6() LT0(Cons(z0,z1),Cons(z2,z3)) -> c5(LT0(z1,z3)) LT0(Nil(),Cons(z0,z1)) -> c4() NOTEMPTY(Cons(z0,z1)) -> c11() NOTEMPTY(Nil()) -> c12() NUMBER4(z0) -> c13() - Weak TRS: F[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c2(F(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c3(F(z0,z2)) G[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c(G(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c1(G(z0,z2)) f(z0,Cons(z1,z2)) -> f[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) f(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) f[Ite][False][Ite](False(),Cons(z0,z1),z2) -> f(z1,Cons(Cons(Nil(),Nil()),z2)) f[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> f(z0,z2) g(z0,Cons(z1,z2)) -> g[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) g(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) g[Ite][False][Ite](False(),Cons(z0,z1),z2) -> g(z1,Cons(Cons(Nil(),Nil()),z2)) g[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> g(z0,z2) goal(z0,z1) -> Cons(f(z0,z1),Cons(g(z0,z1),Nil())) lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() notEmpty(Cons(z0,z1)) -> True() notEmpty(Nil()) -> False() number4(z0) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,c1/1,c10/2,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,F[ITE][FALSE][ITE],G,GOAL,G[ITE][FALSE][ITE],LT0 ,NOTEMPTY,NUMBER4,f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0,notEmpty ,number4} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: F(z0,Cons(z1,z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) F(z0,Nil()) -> c9() G(z0,Cons(z1,z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) G(z0,Nil()) -> c7() GOAL(z0,z1) -> c14(F(z0,z1)) GOAL(z0,z1) -> c15(G(z0,z1)) LT0(z0,Nil()) -> c6() LT0(Cons(z0,z1),Cons(z2,z3)) -> c5(LT0(z1,z3)) LT0(Nil(),Cons(z0,z1)) -> c4() NOTEMPTY(Cons(z0,z1)) -> c11() NOTEMPTY(Nil()) -> c12() NUMBER4(z0) -> c13() - Weak TRS: F[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c2(F(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c3(F(z0,z2)) G[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c(G(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c1(G(z0,z2)) f(z0,Cons(z1,z2)) -> f[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) f(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) f[Ite][False][Ite](False(),Cons(z0,z1),z2) -> f(z1,Cons(Cons(Nil(),Nil()),z2)) f[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> f(z0,z2) g(z0,Cons(z1,z2)) -> g[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) g(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) g[Ite][False][Ite](False(),Cons(z0,z1),z2) -> g(z1,Cons(Cons(Nil(),Nil()),z2)) g[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> g(z0,z2) goal(z0,z1) -> Cons(f(z0,z1),Cons(g(z0,z1),Nil())) lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() notEmpty(Cons(z0,z1)) -> True() notEmpty(Nil()) -> False() number4(z0) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,c1/1,c10/2,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,F[ITE][FALSE][ITE],G,GOAL,G[ITE][FALSE][ITE],LT0 ,NOTEMPTY,NUMBER4,f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0,notEmpty ,number4} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: F(z0,Cons(z1,z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) F(z0,Nil()) -> c9() G(z0,Cons(z1,z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) G(z0,Nil()) -> c7() GOAL(z0,z1) -> c14(F(z0,z1)) GOAL(z0,z1) -> c15(G(z0,z1)) LT0(z0,Nil()) -> c6() LT0(Cons(z0,z1),Cons(z2,z3)) -> c5(LT0(z1,z3)) LT0(Nil(),Cons(z0,z1)) -> c4() NOTEMPTY(Cons(z0,z1)) -> c11() NOTEMPTY(Nil()) -> c12() NUMBER4(z0) -> c13() - Weak TRS: F[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c2(F(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c3(F(z0,z2)) G[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c(G(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c1(G(z0,z2)) f(z0,Cons(z1,z2)) -> f[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) f(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) f[Ite][False][Ite](False(),Cons(z0,z1),z2) -> f(z1,Cons(Cons(Nil(),Nil()),z2)) f[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> f(z0,z2) g(z0,Cons(z1,z2)) -> g[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) g(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) g[Ite][False][Ite](False(),Cons(z0,z1),z2) -> g(z1,Cons(Cons(Nil(),Nil()),z2)) g[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> g(z0,z2) goal(z0,z1) -> Cons(f(z0,z1),Cons(g(z0,z1),Nil())) lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() notEmpty(Cons(z0,z1)) -> True() notEmpty(Nil()) -> False() number4(z0) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,c1/1,c10/2,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,F[ITE][FALSE][ITE],G,GOAL,G[ITE][FALSE][ITE],LT0 ,NOTEMPTY,NUMBER4,f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0,notEmpty ,number4} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: LT0(y,u){y -> Cons(x,y),u -> Cons(z,u)} = LT0(Cons(x,y),Cons(z,u)) ->^+ c5(LT0(y,u)) = C[LT0(y,u) = LT0(y,u){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: F(z0,Cons(z1,z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) F(z0,Nil()) -> c9() G(z0,Cons(z1,z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) G(z0,Nil()) -> c7() GOAL(z0,z1) -> c14(F(z0,z1)) GOAL(z0,z1) -> c15(G(z0,z1)) LT0(z0,Nil()) -> c6() LT0(Cons(z0,z1),Cons(z2,z3)) -> c5(LT0(z1,z3)) LT0(Nil(),Cons(z0,z1)) -> c4() NOTEMPTY(Cons(z0,z1)) -> c11() NOTEMPTY(Nil()) -> c12() NUMBER4(z0) -> c13() - Weak TRS: F[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c2(F(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c3(F(z0,z2)) G[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c(G(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c1(G(z0,z2)) f(z0,Cons(z1,z2)) -> f[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) f(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) f[Ite][False][Ite](False(),Cons(z0,z1),z2) -> f(z1,Cons(Cons(Nil(),Nil()),z2)) f[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> f(z0,z2) g(z0,Cons(z1,z2)) -> g[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) g(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) g[Ite][False][Ite](False(),Cons(z0,z1),z2) -> g(z1,Cons(Cons(Nil(),Nil()),z2)) g[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> g(z0,z2) goal(z0,z1) -> Cons(f(z0,z1),Cons(g(z0,z1),Nil())) lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() notEmpty(Cons(z0,z1)) -> True() notEmpty(Nil()) -> False() number4(z0) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,c1/1,c10/2,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,F[ITE][FALSE][ITE],G,GOAL,G[ITE][FALSE][ITE],LT0 ,NOTEMPTY,NUMBER4,f,f[Ite][False][Ite],g,g[Ite][False][Ite],goal,lt0,notEmpty ,number4} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) F#(z0,Nil()) -> c_2() G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) G#(z0,Nil()) -> c_4() GOAL#(z0,z1) -> c_5(F#(z0,z1)) GOAL#(z0,z1) -> c_6(G#(z0,z1)) LT0#(z0,Nil()) -> c_7() LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) LT0#(Nil(),Cons(z0,z1)) -> c_9() NOTEMPTY#(Cons(z0,z1)) -> c_10() NOTEMPTY#(Nil()) -> c_11() NUMBER4#(z0) -> c_12() Weak DPs F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) f#(z0,Cons(z1,z2)) -> c_17(f[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))) f#(z0,Nil()) -> c_18() f[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_19(f#(z1,Cons(Cons(Nil(),Nil()),z2))) f[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_20(f#(z0,z2)) g#(z0,Cons(z1,z2)) -> c_21(g[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))) g#(z0,Nil()) -> c_22() g[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_23(g#(z1,Cons(Cons(Nil(),Nil()),z2))) g[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_24(g#(z0,z2)) goal#(z0,z1) -> c_25(f#(z0,z1),g#(z0,z1)) lt0#(z0,Nil()) -> c_26() lt0#(Cons(z0,z1),Cons(z2,z3)) -> c_27(lt0#(z1,z3)) lt0#(Nil(),Cons(z0,z1)) -> c_28() notEmpty#(Cons(z0,z1)) -> c_29() notEmpty#(Nil()) -> c_30() number4#(z0) -> c_31() and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) F#(z0,Nil()) -> c_2() G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) G#(z0,Nil()) -> c_4() GOAL#(z0,z1) -> c_5(F#(z0,z1)) GOAL#(z0,z1) -> c_6(G#(z0,z1)) LT0#(z0,Nil()) -> c_7() LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) LT0#(Nil(),Cons(z0,z1)) -> c_9() NOTEMPTY#(Cons(z0,z1)) -> c_10() NOTEMPTY#(Nil()) -> c_11() NUMBER4#(z0) -> c_12() - Weak DPs: F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) f#(z0,Cons(z1,z2)) -> c_17(f[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))) f#(z0,Nil()) -> c_18() f[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_19(f#(z1,Cons(Cons(Nil(),Nil()),z2))) f[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_20(f#(z0,z2)) g#(z0,Cons(z1,z2)) -> c_21(g[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))) g#(z0,Nil()) -> c_22() g[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_23(g#(z1,Cons(Cons(Nil(),Nil()),z2))) g[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_24(g#(z0,z2)) goal#(z0,z1) -> c_25(f#(z0,z1),g#(z0,z1)) lt0#(z0,Nil()) -> c_26() lt0#(Cons(z0,z1),Cons(z2,z3)) -> c_27(lt0#(z1,z3)) lt0#(Nil(),Cons(z0,z1)) -> c_28() notEmpty#(Cons(z0,z1)) -> c_29() notEmpty#(Nil()) -> c_30() number4#(z0) -> c_31() - Weak TRS: F(z0,Cons(z1,z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) F(z0,Nil()) -> c9() F[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c2(F(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c3(F(z0,z2)) G(z0,Cons(z1,z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) G(z0,Nil()) -> c7() GOAL(z0,z1) -> c14(F(z0,z1)) GOAL(z0,z1) -> c15(G(z0,z1)) G[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c(G(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c1(G(z0,z2)) LT0(z0,Nil()) -> c6() LT0(Cons(z0,z1),Cons(z2,z3)) -> c5(LT0(z1,z3)) LT0(Nil(),Cons(z0,z1)) -> c4() NOTEMPTY(Cons(z0,z1)) -> c11() NOTEMPTY(Nil()) -> c12() NUMBER4(z0) -> c13() f(z0,Cons(z1,z2)) -> f[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) f(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) f[Ite][False][Ite](False(),Cons(z0,z1),z2) -> f(z1,Cons(Cons(Nil(),Nil()),z2)) f[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> f(z0,z2) g(z0,Cons(z1,z2)) -> g[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) g(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) g[Ite][False][Ite](False(),Cons(z0,z1),z2) -> g(z1,Cons(Cons(Nil(),Nil()),z2)) g[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> g(z0,z2) goal(z0,z1) -> Cons(f(z0,z1),Cons(g(z0,z1),Nil())) lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() notEmpty(Cons(z0,z1)) -> True() notEmpty(Nil()) -> False() number4(z0) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/3,c_2/0,c_3/3,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {7,9,10,11,12} by application of Pre({7,9,10,11,12}) = {1,3,8}. Here rules are labelled as follows: 1: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) 2: F#(z0,Nil()) -> c_2() 3: G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) 4: G#(z0,Nil()) -> c_4() 5: GOAL#(z0,z1) -> c_5(F#(z0,z1)) 6: GOAL#(z0,z1) -> c_6(G#(z0,z1)) 7: LT0#(z0,Nil()) -> c_7() 8: LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) 9: LT0#(Nil(),Cons(z0,z1)) -> c_9() 10: NOTEMPTY#(Cons(z0,z1)) -> c_10() 11: NOTEMPTY#(Nil()) -> c_11() 12: NUMBER4#(z0) -> c_12() 13: F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) 14: F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) 15: G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) 16: G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) 17: f#(z0,Cons(z1,z2)) -> c_17(f[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))) 18: f#(z0,Nil()) -> c_18() 19: f[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_19(f#(z1,Cons(Cons(Nil(),Nil()),z2))) 20: f[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_20(f#(z0,z2)) 21: g#(z0,Cons(z1,z2)) -> c_21(g[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))) 22: g#(z0,Nil()) -> c_22() 23: g[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_23(g#(z1,Cons(Cons(Nil(),Nil()),z2))) 24: g[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_24(g#(z0,z2)) 25: goal#(z0,z1) -> c_25(f#(z0,z1),g#(z0,z1)) 26: lt0#(z0,Nil()) -> c_26() 27: lt0#(Cons(z0,z1),Cons(z2,z3)) -> c_27(lt0#(z1,z3)) 28: lt0#(Nil(),Cons(z0,z1)) -> c_28() 29: notEmpty#(Cons(z0,z1)) -> c_29() 30: notEmpty#(Nil()) -> c_30() 31: number4#(z0) -> c_31() ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) F#(z0,Nil()) -> c_2() G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) G#(z0,Nil()) -> c_4() GOAL#(z0,z1) -> c_5(F#(z0,z1)) GOAL#(z0,z1) -> c_6(G#(z0,z1)) LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) - Weak DPs: F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) LT0#(z0,Nil()) -> c_7() LT0#(Nil(),Cons(z0,z1)) -> c_9() NOTEMPTY#(Cons(z0,z1)) -> c_10() NOTEMPTY#(Nil()) -> c_11() NUMBER4#(z0) -> c_12() f#(z0,Cons(z1,z2)) -> c_17(f[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))) f#(z0,Nil()) -> c_18() f[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_19(f#(z1,Cons(Cons(Nil(),Nil()),z2))) f[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_20(f#(z0,z2)) g#(z0,Cons(z1,z2)) -> c_21(g[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))) g#(z0,Nil()) -> c_22() g[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_23(g#(z1,Cons(Cons(Nil(),Nil()),z2))) g[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_24(g#(z0,z2)) goal#(z0,z1) -> c_25(f#(z0,z1),g#(z0,z1)) lt0#(z0,Nil()) -> c_26() lt0#(Cons(z0,z1),Cons(z2,z3)) -> c_27(lt0#(z1,z3)) lt0#(Nil(),Cons(z0,z1)) -> c_28() notEmpty#(Cons(z0,z1)) -> c_29() notEmpty#(Nil()) -> c_30() number4#(z0) -> c_31() - Weak TRS: F(z0,Cons(z1,z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) F(z0,Nil()) -> c9() F[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c2(F(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c3(F(z0,z2)) G(z0,Cons(z1,z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) G(z0,Nil()) -> c7() GOAL(z0,z1) -> c14(F(z0,z1)) GOAL(z0,z1) -> c15(G(z0,z1)) G[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c(G(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c1(G(z0,z2)) LT0(z0,Nil()) -> c6() LT0(Cons(z0,z1),Cons(z2,z3)) -> c5(LT0(z1,z3)) LT0(Nil(),Cons(z0,z1)) -> c4() NOTEMPTY(Cons(z0,z1)) -> c11() NOTEMPTY(Nil()) -> c12() NUMBER4(z0) -> c13() f(z0,Cons(z1,z2)) -> f[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) f(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) f[Ite][False][Ite](False(),Cons(z0,z1),z2) -> f(z1,Cons(Cons(Nil(),Nil()),z2)) f[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> f(z0,z2) g(z0,Cons(z1,z2)) -> g[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) g(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) g[Ite][False][Ite](False(),Cons(z0,z1),z2) -> g(z1,Cons(Cons(Nil(),Nil()),z2)) g[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> g(z0,z2) goal(z0,z1) -> Cons(f(z0,z1),Cons(g(z0,z1),Nil())) lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() notEmpty(Cons(z0,z1)) -> True() notEmpty(Nil()) -> False() number4(z0) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/3,c_2/0,c_3/3,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) -->_2 lt0#(Cons(z0,z1),Cons(z2,z3)) -> c_27(lt0#(z1,z3)):27 -->_1 F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)):9 -->_1 F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))):8 -->_3 LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)):7 -->_2 lt0#(Nil(),Cons(z0,z1)) -> c_28():28 -->_3 LT0#(Nil(),Cons(z0,z1)) -> c_9():13 2:S:F#(z0,Nil()) -> c_2() 3:S:G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) -->_2 lt0#(Cons(z0,z1),Cons(z2,z3)) -> c_27(lt0#(z1,z3)):27 -->_1 G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)):11 -->_1 G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))):10 -->_3 LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)):7 -->_2 lt0#(Nil(),Cons(z0,z1)) -> c_28():28 -->_3 LT0#(Nil(),Cons(z0,z1)) -> c_9():13 4:S:G#(z0,Nil()) -> c_4() 5:S:GOAL#(z0,z1) -> c_5(F#(z0,z1)) -->_1 F#(z0,Nil()) -> c_2():2 -->_1 F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))):1 6:S:GOAL#(z0,z1) -> c_6(G#(z0,z1)) -->_1 G#(z0,Nil()) -> c_4():4 -->_1 G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))):3 7:S:LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) -->_1 LT0#(Nil(),Cons(z0,z1)) -> c_9():13 -->_1 LT0#(z0,Nil()) -> c_7():12 -->_1 LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)):7 8:W:F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) -->_1 F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))):1 9:W:F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) -->_1 F#(z0,Nil()) -> c_2():2 -->_1 F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))):1 10:W:G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) -->_1 G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))):3 11:W:G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) -->_1 G#(z0,Nil()) -> c_4():4 -->_1 G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))):3 12:W:LT0#(z0,Nil()) -> c_7() 13:W:LT0#(Nil(),Cons(z0,z1)) -> c_9() 14:W:NOTEMPTY#(Cons(z0,z1)) -> c_10() 15:W:NOTEMPTY#(Nil()) -> c_11() 16:W:NUMBER4#(z0) -> c_12() 17:W:f#(z0,Cons(z1,z2)) -> c_17(f[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))) -->_2 lt0#(Cons(z0,z1),Cons(z2,z3)) -> c_27(lt0#(z1,z3)):27 -->_1 f[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_20(f#(z0,z2)):20 -->_1 f[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_19(f#(z1,Cons(Cons(Nil(),Nil()),z2))):19 -->_2 lt0#(Nil(),Cons(z0,z1)) -> c_28():28 18:W:f#(z0,Nil()) -> c_18() 19:W:f[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_19(f#(z1,Cons(Cons(Nil(),Nil()),z2))) -->_1 f#(z0,Cons(z1,z2)) -> c_17(f[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))):17 20:W:f[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_20(f#(z0,z2)) -->_1 f#(z0,Nil()) -> c_18():18 -->_1 f#(z0,Cons(z1,z2)) -> c_17(f[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))):17 21:W:g#(z0,Cons(z1,z2)) -> c_21(g[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))) -->_2 lt0#(Cons(z0,z1),Cons(z2,z3)) -> c_27(lt0#(z1,z3)):27 -->_1 g[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_24(g#(z0,z2)):24 -->_1 g[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_23(g#(z1,Cons(Cons(Nil(),Nil()),z2))):23 -->_2 lt0#(Nil(),Cons(z0,z1)) -> c_28():28 22:W:g#(z0,Nil()) -> c_22() 23:W:g[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_23(g#(z1,Cons(Cons(Nil(),Nil()),z2))) -->_1 g#(z0,Cons(z1,z2)) -> c_21(g[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))):21 24:W:g[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_24(g#(z0,z2)) -->_1 g#(z0,Nil()) -> c_22():22 -->_1 g#(z0,Cons(z1,z2)) -> c_21(g[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))):21 25:W:goal#(z0,z1) -> c_25(f#(z0,z1),g#(z0,z1)) -->_2 g#(z0,Nil()) -> c_22():22 -->_2 g#(z0,Cons(z1,z2)) -> c_21(g[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))):21 -->_1 f#(z0,Nil()) -> c_18():18 -->_1 f#(z0,Cons(z1,z2)) -> c_17(f[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))):17 26:W:lt0#(z0,Nil()) -> c_26() 27:W:lt0#(Cons(z0,z1),Cons(z2,z3)) -> c_27(lt0#(z1,z3)) -->_1 lt0#(Nil(),Cons(z0,z1)) -> c_28():28 -->_1 lt0#(Cons(z0,z1),Cons(z2,z3)) -> c_27(lt0#(z1,z3)):27 -->_1 lt0#(z0,Nil()) -> c_26():26 28:W:lt0#(Nil(),Cons(z0,z1)) -> c_28() 29:W:notEmpty#(Cons(z0,z1)) -> c_29() 30:W:notEmpty#(Nil()) -> c_30() 31:W:number4#(z0) -> c_31() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 31: number4#(z0) -> c_31() 30: notEmpty#(Nil()) -> c_30() 29: notEmpty#(Cons(z0,z1)) -> c_29() 25: goal#(z0,z1) -> c_25(f#(z0,z1),g#(z0,z1)) 21: g#(z0,Cons(z1,z2)) -> c_21(g[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))) 24: g[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_24(g#(z0,z2)) 23: g[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_23(g#(z1,Cons(Cons(Nil(),Nil()),z2))) 22: g#(z0,Nil()) -> c_22() 17: f#(z0,Cons(z1,z2)) -> c_17(f[Ite][False][Ite]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil()))) 20: f[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_20(f#(z0,z2)) 19: f[Ite][False][Ite]#(False(),Cons(z0,z1),z2) -> c_19(f#(z1,Cons(Cons(Nil(),Nil()),z2))) 18: f#(z0,Nil()) -> c_18() 16: NUMBER4#(z0) -> c_12() 15: NOTEMPTY#(Nil()) -> c_11() 14: NOTEMPTY#(Cons(z0,z1)) -> c_10() 12: LT0#(z0,Nil()) -> c_7() 13: LT0#(Nil(),Cons(z0,z1)) -> c_9() 27: lt0#(Cons(z0,z1),Cons(z2,z3)) -> c_27(lt0#(z1,z3)) 26: lt0#(z0,Nil()) -> c_26() 28: lt0#(Nil(),Cons(z0,z1)) -> c_28() ** Step 1.b:4: SimplifyRHS. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) F#(z0,Nil()) -> c_2() G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) G#(z0,Nil()) -> c_4() GOAL#(z0,z1) -> c_5(F#(z0,z1)) GOAL#(z0,z1) -> c_6(G#(z0,z1)) LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) - Weak DPs: F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) - Weak TRS: F(z0,Cons(z1,z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) F(z0,Nil()) -> c9() F[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c2(F(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c3(F(z0,z2)) G(z0,Cons(z1,z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) G(z0,Nil()) -> c7() GOAL(z0,z1) -> c14(F(z0,z1)) GOAL(z0,z1) -> c15(G(z0,z1)) G[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c(G(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c1(G(z0,z2)) LT0(z0,Nil()) -> c6() LT0(Cons(z0,z1),Cons(z2,z3)) -> c5(LT0(z1,z3)) LT0(Nil(),Cons(z0,z1)) -> c4() NOTEMPTY(Cons(z0,z1)) -> c11() NOTEMPTY(Nil()) -> c12() NUMBER4(z0) -> c13() f(z0,Cons(z1,z2)) -> f[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) f(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) f[Ite][False][Ite](False(),Cons(z0,z1),z2) -> f(z1,Cons(Cons(Nil(),Nil()),z2)) f[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> f(z0,z2) g(z0,Cons(z1,z2)) -> g[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) g(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) g[Ite][False][Ite](False(),Cons(z0,z1),z2) -> g(z1,Cons(Cons(Nil(),Nil()),z2)) g[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> g(z0,z2) goal(z0,z1) -> Cons(f(z0,z1),Cons(g(z0,z1),Nil())) lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() notEmpty(Cons(z0,z1)) -> True() notEmpty(Nil()) -> False() number4(z0) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/3,c_2/0,c_3/3,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) -->_1 F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)):9 -->_1 F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))):8 -->_3 LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)):7 2:S:F#(z0,Nil()) -> c_2() 3:S:G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))) -->_1 G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)):11 -->_1 G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))):10 -->_3 LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)):7 4:S:G#(z0,Nil()) -> c_4() 5:S:GOAL#(z0,z1) -> c_5(F#(z0,z1)) -->_1 F#(z0,Nil()) -> c_2():2 -->_1 F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))):1 6:S:GOAL#(z0,z1) -> c_6(G#(z0,z1)) -->_1 G#(z0,Nil()) -> c_4():4 -->_1 G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))):3 7:S:LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) -->_1 LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)):7 8:W:F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) -->_1 F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))):1 9:W:F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) -->_1 F#(z0,Nil()) -> c_2():2 -->_1 F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))):1 10:W:G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) -->_1 G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))):3 11:W:G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) -->_1 G#(z0,Nil()) -> c_4():4 -->_1 G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,lt0#(z0,Cons(Nil(),Nil())) ,LT0#(z0,Cons(Nil(),Nil()))):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) ** Step 1.b:5: UsableRules. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) F#(z0,Nil()) -> c_2() G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) G#(z0,Nil()) -> c_4() GOAL#(z0,z1) -> c_5(F#(z0,z1)) GOAL#(z0,z1) -> c_6(G#(z0,z1)) LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) - Weak DPs: F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) - Weak TRS: F(z0,Cons(z1,z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) F(z0,Nil()) -> c9() F[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c2(F(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c3(F(z0,z2)) G(z0,Cons(z1,z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0(z0,Cons(Nil(),Nil()))) G(z0,Nil()) -> c7() GOAL(z0,z1) -> c14(F(z0,z1)) GOAL(z0,z1) -> c15(G(z0,z1)) G[ITE][FALSE][ITE](False(),Cons(z0,z1),z2) -> c(G(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c1(G(z0,z2)) LT0(z0,Nil()) -> c6() LT0(Cons(z0,z1),Cons(z2,z3)) -> c5(LT0(z1,z3)) LT0(Nil(),Cons(z0,z1)) -> c4() NOTEMPTY(Cons(z0,z1)) -> c11() NOTEMPTY(Nil()) -> c12() NUMBER4(z0) -> c13() f(z0,Cons(z1,z2)) -> f[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) f(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) f[Ite][False][Ite](False(),Cons(z0,z1),z2) -> f(z1,Cons(Cons(Nil(),Nil()),z2)) f[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> f(z0,z2) g(z0,Cons(z1,z2)) -> g[Ite][False][Ite](lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) g(z0,Nil()) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) g[Ite][False][Ite](False(),Cons(z0,z1),z2) -> g(z1,Cons(Cons(Nil(),Nil()),z2)) g[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> g(z0,z2) goal(z0,z1) -> Cons(f(z0,z1),Cons(g(z0,z1),Nil())) lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() notEmpty(Cons(z0,z1)) -> True() notEmpty(Nil()) -> False() number4(z0) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil())))) - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) F#(z0,Nil()) -> c_2() F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) G#(z0,Nil()) -> c_4() GOAL#(z0,z1) -> c_5(F#(z0,z1)) GOAL#(z0,z1) -> c_6(G#(z0,z1)) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) ** Step 1.b:6: RemoveHeads. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) F#(z0,Nil()) -> c_2() G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) G#(z0,Nil()) -> c_4() GOAL#(z0,z1) -> c_5(F#(z0,z1)) GOAL#(z0,z1) -> c_6(G#(z0,z1)) LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) - Weak DPs: F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) - Weak TRS: lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) -->_1 F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)):9 -->_1 F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))):8 -->_2 LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)):7 2:S:F#(z0,Nil()) -> c_2() 3:S:G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) -->_1 G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)):11 -->_1 G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))):10 -->_2 LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)):7 4:S:G#(z0,Nil()) -> c_4() 5:S:GOAL#(z0,z1) -> c_5(F#(z0,z1)) -->_1 F#(z0,Nil()) -> c_2():2 -->_1 F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))):1 6:S:GOAL#(z0,z1) -> c_6(G#(z0,z1)) -->_1 G#(z0,Nil()) -> c_4():4 -->_1 G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))):3 7:S:LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) -->_1 LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)):7 8:W:F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) -->_1 F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))):1 9:W:F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) -->_1 F#(z0,Nil()) -> c_2():2 -->_1 F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))):1 10:W:G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) -->_1 G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))):3 11:W:G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) -->_1 G#(z0,Nil()) -> c_4():4 -->_1 G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,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#(z0,z1) -> c_5(F#(z0,z1))),(6,GOAL#(z0,z1) -> c_6(G#(z0,z1)))] ** Step 1.b:7: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) F#(z0,Nil()) -> c_2() G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) G#(z0,Nil()) -> c_4() LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) - Weak DPs: F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) - Weak TRS: lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + 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#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) and a lower component F#(z0,Nil()) -> c_2() G#(z0,Nil()) -> c_4() LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) Further, following extension rules are added to the lower component. F#(z0,Cons(z1,z2)) -> F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) F#(z0,Cons(z1,z2)) -> LT0#(z0,Cons(Nil(),Nil())) F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> F#(z1,Cons(Cons(Nil(),Nil()),z2)) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> F#(z0,z2) G#(z0,Cons(z1,z2)) -> G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) G#(z0,Cons(z1,z2)) -> LT0#(z0,Cons(Nil(),Nil())) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> G#(z1,Cons(Cons(Nil(),Nil()),z2)) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> G#(z0,z2) *** Step 1.b:7.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) - Weak DPs: F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) - Weak TRS: lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) -->_1 F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))):5 -->_1 F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)):2 2:S:F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) -->_1 F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))):1 3:S:G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))) -->_1 G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))):6 -->_1 G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)):4 4:S:G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) -->_1 G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))):3 5:W:F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) -->_1 F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))):1 6:W:G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) -->_1 G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) ,LT0#(z0,Cons(Nil(),Nil()))):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) *** Step 1.b:7.a:2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) - Weak DPs: F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) - Weak TRS: lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + 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_13) = {1}, uargs(c_14) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(Cons) = [0] p(F) = [0] p(F[ITE][FALSE][ITE]) = [0] p(False) = [6] p(G) = [0] p(GOAL) = [0] p(G[ITE][FALSE][ITE]) = [0] p(LT0) = [0] p(NOTEMPTY) = [0] p(NUMBER4) = [0] p(Nil) = [0] p(True) = [0] p(c) = [1] x1 + [0] p(c1) = [1] x1 + [0] p(c10) = [1] x1 + [1] x2 + [0] p(c11) = [0] p(c12) = [0] p(c13) = [0] p(c14) = [1] x1 + [0] p(c15) = [1] x1 + [0] p(c2) = [1] x1 + [0] p(c3) = [1] x1 + [0] p(c4) = [0] p(c5) = [1] x1 + [0] p(c6) = [0] p(c7) = [0] p(c8) = [1] x1 + [1] x2 + [0] p(c9) = [0] p(f) = [0] p(f[Ite][False][Ite]) = [0] p(g) = [0] p(g[Ite][False][Ite]) = [0] p(goal) = [1] x1 + [0] p(lt0) = [6] p(notEmpty) = [1] x1 + [2] p(number4) = [2] x1 + [1] p(F#) = [3] p(F[ITE][FALSE][ITE]#) = [1] x1 + [4] x3 + [4] p(G#) = [3] p(GOAL#) = [0] p(G[ITE][FALSE][ITE]#) = [1] x1 + [0] p(LT0#) = [0] p(NOTEMPTY#) = [0] p(NUMBER4#) = [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(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [1] p(c_14) = [1] x1 + [0] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [2] p(c_27) = [1] x1 + [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [4] Following rules are strictly oriented: F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) = [4] > [3] = c_14(F#(z0,z2)) Following rules are (at-least) weakly oriented: F#(z0,Cons(z1,z2)) = [3] >= [10] = c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) = [4] z2 + [10] >= [4] = c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) G#(z0,Cons(z1,z2)) = [3] >= [6] = c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) = [6] >= [3] = c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) = [0] >= [3] = c_16(G#(z0,z2)) lt0(z0,Nil()) = [6] >= [6] = False() lt0(Cons(z0,z1),Cons(z2,z3)) = [6] >= [6] = lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) = [6] >= [0] = True() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:7.a:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) - Weak DPs: F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) - Weak TRS: lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_3) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0#,NOTEMPTY#,NUMBER4#,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(F) = [1] p(F[ITE][FALSE][ITE]) = [4] x1 + [1] x2 + [1] p(False) = [1] p(G) = [1] x2 + [2] p(GOAL) = [1] p(G[ITE][FALSE][ITE]) = [1] x1 + [1] x3 + [0] p(LT0) = [1] x1 + [2] p(NOTEMPTY) = [1] p(NUMBER4) = [1] x1 + [2] p(Nil) = [14] p(True) = [0] p(c) = [1] x1 + [2] p(c1) = [1] p(c10) = [1] x1 + [0] p(c11) = [2] p(c12) = [0] p(c13) = [0] p(c14) = [0] p(c15) = [8] p(c2) = [1] x1 + [1] p(c3) = [2] p(c4) = [2] p(c5) = [2] p(c6) = [1] p(c7) = [1] p(c8) = [1] p(c9) = [1] p(f) = [4] x1 + [1] x2 + [0] p(f[Ite][False][Ite]) = [1] x1 + [1] x2 + [1] p(g) = [1] x2 + [1] p(g[Ite][False][Ite]) = [1] x1 + [1] x2 + [0] p(goal) = [1] x1 + [1] x2 + [0] p(lt0) = [8] p(notEmpty) = [0] p(number4) = [1] x1 + [0] p(F#) = [10] x1 + [8] x2 + [15] p(F[ITE][FALSE][ITE]#) = [10] x2 + [8] x3 + [11] p(G#) = [0] p(GOAL#) = [2] x1 + [1] x2 + [0] p(G[ITE][FALSE][ITE]#) = [0] p(LT0#) = [2] x1 + [1] x2 + [2] p(NOTEMPTY#) = [1] x1 + [0] p(NUMBER4#) = [1] x1 + [8] p(f#) = [2] x1 + [8] x2 + [2] p(f[Ite][False][Ite]#) = [1] x2 + [8] x3 + [1] p(g#) = [1] p(g[Ite][False][Ite]#) = [1] x1 + [1] x3 + [8] p(goal#) = [1] x2 + [1] p(lt0#) = [8] p(notEmpty#) = [1] x1 + [0] p(number4#) = [2] x1 + [1] p(c_1) = [1] x1 + [2] p(c_2) = [2] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [2] x1 + [0] p(c_6) = [1] p(c_7) = [1] p(c_8) = [2] p(c_9) = [0] p(c_10) = [1] p(c_11) = [0] p(c_12) = [2] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [2] p(c_15) = [8] x1 + [0] p(c_16) = [8] x1 + [0] p(c_17) = [1] x1 + [0] p(c_18) = [1] p(c_19) = [2] x1 + [1] p(c_20) = [4] p(c_21) = [1] x1 + [0] p(c_22) = [1] p(c_23) = [2] x1 + [1] p(c_24) = [1] p(c_25) = [1] x1 + [1] x2 + [0] p(c_26) = [0] p(c_27) = [1] p(c_28) = [8] p(c_29) = [1] p(c_30) = [1] p(c_31) = [0] Following rules are strictly oriented: F#(z0,Cons(z1,z2)) = [10] z0 + [8] z2 + [31] > [10] z0 + [8] z2 + [29] = c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) Following rules are (at-least) weakly oriented: F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) = [10] z1 + [8] z2 + [31] >= [10] z1 + [8] z2 + [31] = c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) = [10] z0 + [8] z2 + [27] >= [10] z0 + [8] z2 + [17] = c_14(F#(z0,z2)) G#(z0,Cons(z1,z2)) = [0] >= [0] = c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) = [0] >= [0] = c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) = [0] >= [0] = c_16(G#(z0,z2)) *** Step 1.b:7.a:4: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) - Weak DPs: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) - Weak TRS: lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_3) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0#,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g# ,g[Ite][False][Ite]#,goal#,lt0#,notEmpty#,number4#} TcT has computed the following interpretation: p(Cons) = [1] x2 + [4] p(F) = [1] x1 + [0] p(F[ITE][FALSE][ITE]) = [2] x3 + [1] p(False) = [0] p(G) = [1] x1 + [0] p(GOAL) = [1] x1 + [4] p(G[ITE][FALSE][ITE]) = [1] p(LT0) = [1] x1 + [1] x2 + [2] p(NOTEMPTY) = [8] x1 + [2] p(NUMBER4) = [2] x1 + [0] p(Nil) = [2] p(True) = [0] p(c) = [1] p(c1) = [1] x1 + [0] p(c10) = [1] p(c11) = [1] p(c12) = [0] p(c13) = [1] p(c14) = [8] p(c15) = [8] p(c2) = [2] p(c3) = [1] p(c4) = [0] p(c5) = [1] x1 + [1] p(c6) = [1] p(c7) = [1] p(c8) = [2] p(c9) = [1] p(f) = [1] p(f[Ite][False][Ite]) = [8] x1 + [1] x3 + [1] p(g) = [1] x1 + [2] x2 + [2] p(g[Ite][False][Ite]) = [2] p(goal) = [1] x1 + [1] p(lt0) = [1] x1 + [0] p(notEmpty) = [1] p(number4) = [1] x1 + [4] p(F#) = [0] p(F[ITE][FALSE][ITE]#) = [0] p(G#) = [6] x1 + [4] x2 + [1] p(GOAL#) = [2] x2 + [0] p(G[ITE][FALSE][ITE]#) = [6] x2 + [4] x3 + [0] p(LT0#) = [0] p(NOTEMPTY#) = [1] p(NUMBER4#) = [1] p(f#) = [1] p(f[Ite][False][Ite]#) = [1] x3 + [2] p(g#) = [1] x2 + [1] p(g[Ite][False][Ite]#) = [1] x1 + [0] p(goal#) = [1] x1 + [2] x2 + [1] p(lt0#) = [1] x2 + [0] p(notEmpty#) = [4] x1 + [0] p(number4#) = [1] x1 + [2] p(c_1) = [8] x1 + [0] p(c_2) = [4] p(c_3) = [1] x1 + [0] p(c_4) = [2] p(c_5) = [2] p(c_6) = [2] x1 + [0] p(c_7) = [1] p(c_8) = [4] p(c_9) = [2] p(c_10) = [1] p(c_11) = [2] p(c_12) = [1] p(c_13) = [1] x1 + [0] p(c_14) = [8] x1 + [0] p(c_15) = [1] x1 + [7] p(c_16) = [1] x1 + [7] p(c_17) = [2] x2 + [1] p(c_18) = [2] p(c_19) = [0] p(c_20) = [0] p(c_21) = [1] x1 + [1] x2 + [2] p(c_22) = [0] p(c_23) = [1] x1 + [0] p(c_24) = [2] x1 + [1] p(c_25) = [1] x2 + [1] p(c_26) = [0] p(c_27) = [1] p(c_28) = [2] p(c_29) = [0] p(c_30) = [0] p(c_31) = [1] Following rules are strictly oriented: G#(z0,Cons(z1,z2)) = [6] z0 + [4] z2 + [17] > [6] z0 + [4] z2 + [16] = c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) = [6] z0 + [4] z2 + [16] > [6] z0 + [4] z2 + [8] = c_16(G#(z0,z2)) Following rules are (at-least) weakly oriented: F#(z0,Cons(z1,z2)) = [0] >= [0] = c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) = [0] >= [0] = c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) = [0] >= [0] = c_14(F#(z0,z2)) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) = [6] z1 + [4] z2 + [24] >= [6] z1 + [4] z2 + [24] = c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) *** Step 1.b:7.a:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: F#(z0,Cons(z1,z2)) -> c_1(F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_13(F#(z1,Cons(Cons(Nil(),Nil()),z2))) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_14(F#(z0,z2)) G#(z0,Cons(z1,z2)) -> c_3(G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2))) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> c_15(G#(z1,Cons(Cons(Nil(),Nil()),z2))) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_16(G#(z0,z2)) - Weak TRS: lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:7.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(z0,Nil()) -> c_2() G#(z0,Nil()) -> c_4() LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) - Weak DPs: F#(z0,Cons(z1,z2)) -> F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) F#(z0,Cons(z1,z2)) -> LT0#(z0,Cons(Nil(),Nil())) F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> F#(z1,Cons(Cons(Nil(),Nil()),z2)) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> F#(z0,z2) G#(z0,Cons(z1,z2)) -> G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) G#(z0,Cons(z1,z2)) -> LT0#(z0,Cons(Nil(),Nil())) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> G#(z1,Cons(Cons(Nil(),Nil()),z2)) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> G#(z0,z2) - Weak TRS: lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_8) = {1} Following symbols are considered usable: {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0#,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g# ,g[Ite][False][Ite]#,goal#,lt0#,notEmpty#,number4#} TcT has computed the following interpretation: p(Cons) = [4] p(F) = [0] p(F[ITE][FALSE][ITE]) = [0] p(False) = [0] p(G) = [1] x1 + [0] p(GOAL) = [0] p(G[ITE][FALSE][ITE]) = [0] p(LT0) = [0] p(NOTEMPTY) = [0] p(NUMBER4) = [0] p(Nil) = [0] p(True) = [0] p(c) = [1] x1 + [0] p(c1) = [1] x1 + [0] p(c10) = [1] x1 + [1] x2 + [0] p(c11) = [0] p(c12) = [0] p(c13) = [0] p(c14) = [1] x1 + [0] p(c15) = [1] x1 + [0] p(c2) = [1] x1 + [0] p(c3) = [1] x1 + [0] p(c4) = [0] p(c5) = [1] x1 + [0] p(c6) = [0] p(c7) = [0] p(c8) = [1] x1 + [1] x2 + [0] p(c9) = [0] p(f) = [0] p(f[Ite][False][Ite]) = [1] x2 + [0] p(g) = [1] x1 + [0] p(g[Ite][False][Ite]) = [1] x2 + [1] x3 + [0] p(goal) = [1] x2 + [0] p(lt0) = [1] x2 + [2] p(notEmpty) = [1] x1 + [1] p(number4) = [0] p(F#) = [13] p(F[ITE][FALSE][ITE]#) = [13] p(G#) = [8] p(GOAL#) = [1] x1 + [0] p(G[ITE][FALSE][ITE]#) = [8] p(LT0#) = [0] p(NOTEMPTY#) = [0] p(NUMBER4#) = [4] x1 + [0] p(f#) = [1] x1 + [4] x2 + [0] p(f[Ite][False][Ite]#) = [2] x1 + [1] x2 + [1] x3 + [0] p(g#) = [1] x1 + [1] x2 + [0] p(g[Ite][False][Ite]#) = [1] x1 + [4] x2 + [2] x3 + [0] p(goal#) = [0] p(lt0#) = [1] x2 + [0] p(notEmpty#) = [0] p(number4#) = [1] x1 + [0] p(c_1) = [2] x1 + [1] x2 + [0] p(c_2) = [12] p(c_3) = [1] x1 + [8] x2 + [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [8] x1 + [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [2] p(c_12) = [0] p(c_13) = [1] x1 + [1] p(c_14) = [0] p(c_15) = [1] p(c_16) = [4] x1 + [1] p(c_17) = [2] p(c_18) = [1] p(c_19) = [2] p(c_20) = [1] p(c_21) = [2] x1 + [1] p(c_22) = [1] p(c_23) = [1] x1 + [0] p(c_24) = [2] x1 + [4] p(c_25) = [4] x1 + [1] x2 + [0] p(c_26) = [1] p(c_27) = [1] x1 + [4] p(c_28) = [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [0] Following rules are strictly oriented: F#(z0,Nil()) = [13] > [12] = c_2() G#(z0,Nil()) = [8] > [0] = c_4() Following rules are (at-least) weakly oriented: F#(z0,Cons(z1,z2)) = [13] >= [13] = F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) F#(z0,Cons(z1,z2)) = [13] >= [0] = LT0#(z0,Cons(Nil(),Nil())) F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) = [13] >= [13] = F#(z1,Cons(Cons(Nil(),Nil()),z2)) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) = [13] >= [13] = F#(z0,z2) G#(z0,Cons(z1,z2)) = [8] >= [8] = G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) G#(z0,Cons(z1,z2)) = [8] >= [0] = LT0#(z0,Cons(Nil(),Nil())) G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) = [8] >= [8] = G#(z1,Cons(Cons(Nil(),Nil()),z2)) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) = [8] >= [8] = G#(z0,z2) LT0#(Cons(z0,z1),Cons(z2,z3)) = [0] >= [0] = c_8(LT0#(z1,z3)) *** Step 1.b:7.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) - Weak DPs: F#(z0,Cons(z1,z2)) -> F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) F#(z0,Cons(z1,z2)) -> LT0#(z0,Cons(Nil(),Nil())) F#(z0,Nil()) -> c_2() F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> F#(z1,Cons(Cons(Nil(),Nil()),z2)) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> F#(z0,z2) G#(z0,Cons(z1,z2)) -> G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) G#(z0,Cons(z1,z2)) -> LT0#(z0,Cons(Nil(),Nil())) G#(z0,Nil()) -> c_4() G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> G#(z1,Cons(Cons(Nil(),Nil()),z2)) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> G#(z0,z2) - Weak TRS: lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_8) = {1} Following symbols are considered usable: {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0#,NOTEMPTY#,NUMBER4#,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 + [1] p(F) = [0] p(F[ITE][FALSE][ITE]) = [0] p(False) = [0] p(G) = [0] p(GOAL) = [0] p(G[ITE][FALSE][ITE]) = [0] p(LT0) = [0] p(NOTEMPTY) = [0] p(NUMBER4) = [0] p(Nil) = [0] p(True) = [0] p(c) = [1] x1 + [0] p(c1) = [1] x1 + [0] p(c10) = [1] x1 + [1] x2 + [0] p(c11) = [0] p(c12) = [0] p(c13) = [0] p(c14) = [1] x1 + [0] p(c15) = [1] x1 + [0] p(c2) = [1] x1 + [0] p(c3) = [1] x1 + [0] p(c4) = [0] p(c5) = [1] x1 + [0] p(c6) = [0] p(c7) = [0] p(c8) = [1] x1 + [1] x2 + [0] p(c9) = [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#) = [4] x1 + [0] p(F[ITE][FALSE][ITE]#) = [4] x2 + [0] p(G#) = [4] x1 + [1] x2 + [0] p(GOAL#) = [0] p(G[ITE][FALSE][ITE]#) = [4] x2 + [1] x3 + [0] p(LT0#) = [4] x1 + [0] p(NOTEMPTY#) = [0] p(NUMBER4#) = [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(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] x1 + [2] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [8] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [8] x1 + [1] x2 + [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [2] x1 + [0] p(c_21) = [1] x2 + [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [2] x1 + [0] p(c_25) = [1] x2 + [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [0] Following rules are strictly oriented: LT0#(Cons(z0,z1),Cons(z2,z3)) = [4] z0 + [4] z1 + [4] > [4] z1 + [2] = c_8(LT0#(z1,z3)) Following rules are (at-least) weakly oriented: F#(z0,Cons(z1,z2)) = [4] z0 + [0] >= [4] z0 + [0] = F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) F#(z0,Cons(z1,z2)) = [4] z0 + [0] >= [4] z0 + [0] = LT0#(z0,Cons(Nil(),Nil())) F#(z0,Nil()) = [4] z0 + [0] >= [0] = c_2() F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) = [4] z0 + [4] z1 + [4] >= [4] z1 + [0] = F#(z1,Cons(Cons(Nil(),Nil()),z2)) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) = [4] z0 + [0] >= [4] z0 + [0] = F#(z0,z2) G#(z0,Cons(z1,z2)) = [4] z0 + [1] z1 + [1] z2 + [1] >= [4] z0 + [1] z1 + [1] z2 + [1] = G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) G#(z0,Cons(z1,z2)) = [4] z0 + [1] z1 + [1] z2 + [1] >= [4] z0 + [0] = LT0#(z0,Cons(Nil(),Nil())) G#(z0,Nil()) = [4] z0 + [0] >= [0] = c_4() G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) = [4] z0 + [4] z1 + [1] z2 + [4] >= [4] z1 + [1] z2 + [2] = G#(z1,Cons(Cons(Nil(),Nil()),z2)) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) = [4] z0 + [1] z1 + [1] z2 + [1] >= [4] z0 + [1] z2 + [0] = G#(z0,z2) *** Step 1.b:7.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: F#(z0,Cons(z1,z2)) -> F[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) F#(z0,Cons(z1,z2)) -> LT0#(z0,Cons(Nil(),Nil())) F#(z0,Nil()) -> c_2() F[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> F#(z1,Cons(Cons(Nil(),Nil()),z2)) F[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> F#(z0,z2) G#(z0,Cons(z1,z2)) -> G[ITE][FALSE][ITE]#(lt0(z0,Cons(Nil(),Nil())),z0,Cons(z1,z2)) G#(z0,Cons(z1,z2)) -> LT0#(z0,Cons(Nil(),Nil())) G#(z0,Nil()) -> c_4() G[ITE][FALSE][ITE]#(False(),Cons(z0,z1),z2) -> G#(z1,Cons(Cons(Nil(),Nil()),z2)) G[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> G#(z0,z2) LT0#(Cons(z0,z1),Cons(z2,z3)) -> c_8(LT0#(z1,z3)) - Weak TRS: lt0(z0,Nil()) -> False() lt0(Cons(z0,z1),Cons(z2,z3)) -> lt0(z1,z3) lt0(Nil(),Cons(z0,z1)) -> True() - Signature: {F/2,F[ITE][FALSE][ITE]/3,G/2,GOAL/2,G[ITE][FALSE][ITE]/3,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,F#/2,F[ITE][FALSE][ITE]#/3 ,G#/2,GOAL#/2,G[ITE][FALSE][ITE]#/3,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,c1/1,c10/2 ,c11/0,c12/0,c13/0,c14/1,c15/1,c2/1,c3/1,c4/0,c5/1,c6/0,c7/0,c8/2,c9/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/1,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/1,c_15/1,c_16/1,c_17/2,c_18/0,c_19/1,c_20/1,c_21/2 ,c_22/0,c_23/1,c_24/1,c_25/2,c_26/0,c_27/1,c_28/0,c_29/0,c_30/0,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,F[ITE][FALSE][ITE]#,G#,GOAL#,G[ITE][FALSE][ITE]#,LT0# ,NOTEMPTY#,NUMBER4#,f#,f[Ite][False][Ite]#,g#,g[Ite][False][Ite]#,goal#,lt0#,notEmpty# ,number4#} and constructors {Cons,False,Nil,True,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^2))