MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: a__U11(X) -> U11(X) a__U11(tt()) -> a__U12(tt()) a__U12(X) -> U12(X) a__U12(tt()) -> tt() a____(X,nil()) -> mark(X) a____(X1,X2) -> __(X1,X2) a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(nil(),X) -> mark(X) a__isNePal(X) -> isNePal(X) a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() - Signature: {a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1} / {U11/1,U12/1,__/2,isNePal/1,nil/0,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U12,a____,a__isNePal ,mark} and constructors {U11,U12,__,isNePal,nil,tt} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__U11#(X) -> c_1() a__U11#(tt()) -> c_2(a__U12#(tt())) a__U12#(X) -> c_3() a__U12#(tt()) -> c_4() a____#(X,nil()) -> c_5(mark#(X)) a____#(X1,X2) -> c_6() a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)) a____#(nil(),X) -> c_8(mark#(X)) a__isNePal#(X) -> c_9() a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt())) mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)) mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)) mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)) mark#(nil()) -> c_15() mark#(tt()) -> c_16() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: a__U11#(X) -> c_1() a__U11#(tt()) -> c_2(a__U12#(tt())) a__U12#(X) -> c_3() a__U12#(tt()) -> c_4() a____#(X,nil()) -> c_5(mark#(X)) a____#(X1,X2) -> c_6() a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)) a____#(nil(),X) -> c_8(mark#(X)) a__isNePal#(X) -> c_9() a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt())) mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)) mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)) mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)) mark#(nil()) -> c_15() mark#(tt()) -> c_16() - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> a__U12(tt()) a__U12(X) -> U12(X) a__U12(tt()) -> tt() a____(X,nil()) -> mark(X) a____(X1,X2) -> __(X1,X2) a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(nil(),X) -> mark(X) a__isNePal(X) -> isNePal(X) a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() - Signature: {a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1 ,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/2,c_12/2 ,c_13/3,c_14/2,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal# ,mark#} and constructors {U11,U12,__,isNePal,nil,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,4,6,9,15,16} by application of Pre({1,3,4,6,9,15,16}) = {2,5,7,8,10,11,12,13,14}. Here rules are labelled as follows: 1: a__U11#(X) -> c_1() 2: a__U11#(tt()) -> c_2(a__U12#(tt())) 3: a__U12#(X) -> c_3() 4: a__U12#(tt()) -> c_4() 5: a____#(X,nil()) -> c_5(mark#(X)) 6: a____#(X1,X2) -> c_6() 7: a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)) 8: a____#(nil(),X) -> c_8(mark#(X)) 9: a__isNePal#(X) -> c_9() 10: a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt())) 11: mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)) 12: mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)) 13: mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) 14: mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)) 15: mark#(nil()) -> c_15() 16: mark#(tt()) -> c_16() * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: a__U11#(tt()) -> c_2(a__U12#(tt())) a____#(X,nil()) -> c_5(mark#(X)) a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)) a____#(nil(),X) -> c_8(mark#(X)) a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt())) mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)) mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)) mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)) - Weak DPs: a__U11#(X) -> c_1() a__U12#(X) -> c_3() a__U12#(tt()) -> c_4() a____#(X1,X2) -> c_6() a__isNePal#(X) -> c_9() mark#(nil()) -> c_15() mark#(tt()) -> c_16() - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> a__U12(tt()) a__U12(X) -> U12(X) a__U12(tt()) -> tt() a____(X,nil()) -> mark(X) a____(X1,X2) -> __(X1,X2) a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(nil(),X) -> mark(X) a__isNePal(X) -> isNePal(X) a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() - Signature: {a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1 ,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/2,c_12/2 ,c_13/3,c_14/2,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal# ,mark#} and constructors {U11,U12,__,isNePal,nil,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {5,6}. Here rules are labelled as follows: 1: a__U11#(tt()) -> c_2(a__U12#(tt())) 2: a____#(X,nil()) -> c_5(mark#(X)) 3: a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)) 4: a____#(nil(),X) -> c_8(mark#(X)) 5: a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt())) 6: mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)) 7: mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)) 8: mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) 9: mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)) 10: a__U11#(X) -> c_1() 11: a__U12#(X) -> c_3() 12: a__U12#(tt()) -> c_4() 13: a____#(X1,X2) -> c_6() 14: a__isNePal#(X) -> c_9() 15: mark#(nil()) -> c_15() 16: mark#(tt()) -> c_16() * Step 4: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: a____#(X,nil()) -> c_5(mark#(X)) a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)) a____#(nil(),X) -> c_8(mark#(X)) a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt())) mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)) mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)) mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)) - Weak DPs: a__U11#(X) -> c_1() a__U11#(tt()) -> c_2(a__U12#(tt())) a__U12#(X) -> c_3() a__U12#(tt()) -> c_4() a____#(X1,X2) -> c_6() a__isNePal#(X) -> c_9() mark#(nil()) -> c_15() mark#(tt()) -> c_16() - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> a__U12(tt()) a__U12(X) -> U12(X) a__U12(tt()) -> tt() a____(X,nil()) -> mark(X) a____(X1,X2) -> __(X1,X2) a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(nil(),X) -> mark(X) a__isNePal(X) -> isNePal(X) a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() - Signature: {a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1 ,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/2,c_12/2 ,c_13/3,c_14/2,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal# ,mark#} and constructors {U11,U12,__,isNePal,nil,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4} by application of Pre({4}) = {8}. Here rules are labelled as follows: 1: a____#(X,nil()) -> c_5(mark#(X)) 2: a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)) 3: a____#(nil(),X) -> c_8(mark#(X)) 4: a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt())) 5: mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)) 6: mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)) 7: mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) 8: mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)) 9: a__U11#(X) -> c_1() 10: a__U11#(tt()) -> c_2(a__U12#(tt())) 11: a__U12#(X) -> c_3() 12: a__U12#(tt()) -> c_4() 13: a____#(X1,X2) -> c_6() 14: a__isNePal#(X) -> c_9() 15: mark#(nil()) -> c_15() 16: mark#(tt()) -> c_16() * Step 5: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: a____#(X,nil()) -> c_5(mark#(X)) a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)) a____#(nil(),X) -> c_8(mark#(X)) mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)) mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)) mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)) - Weak DPs: a__U11#(X) -> c_1() a__U11#(tt()) -> c_2(a__U12#(tt())) a__U12#(X) -> c_3() a__U12#(tt()) -> c_4() a____#(X1,X2) -> c_6() a__isNePal#(X) -> c_9() a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt())) mark#(nil()) -> c_15() mark#(tt()) -> c_16() - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> a__U12(tt()) a__U12(X) -> U12(X) a__U12(tt()) -> tt() a____(X,nil()) -> mark(X) a____(X1,X2) -> __(X1,X2) a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(nil(),X) -> mark(X) a__isNePal(X) -> isNePal(X) a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() - Signature: {a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1 ,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/2,c_12/2 ,c_13/3,c_14/2,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal# ,mark#} and constructors {U11,U12,__,isNePal,nil,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a____#(X,nil()) -> c_5(mark#(X)) -->_1 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_1 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_1 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_1 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 -->_1 mark#(tt()) -> c_16():16 -->_1 mark#(nil()) -> c_15():15 2:S:a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)) -->_5 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_4 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_5 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_4 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_5 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_4 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_5 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 -->_4 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 -->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 -->_3 a____#(nil(),X) -> c_8(mark#(X)):3 -->_1 a____#(nil(),X) -> c_8(mark#(X)):3 -->_5 mark#(tt()) -> c_16():16 -->_4 mark#(tt()) -> c_16():16 -->_2 mark#(tt()) -> c_16():16 -->_5 mark#(nil()) -> c_15():15 -->_4 mark#(nil()) -> c_15():15 -->_2 mark#(nil()) -> c_15():15 -->_3 a____#(X1,X2) -> c_6():12 -->_1 a____#(X1,X2) -> c_6():12 -->_3 a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)):2 -->_1 a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)):2 -->_3 a____#(X,nil()) -> c_5(mark#(X)):1 -->_1 a____#(X,nil()) -> c_5(mark#(X)):1 3:S:a____#(nil(),X) -> c_8(mark#(X)) -->_1 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_1 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_1 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_1 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 -->_1 mark#(tt()) -> c_16():16 -->_1 mark#(nil()) -> c_15():15 4:S:mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)) -->_1 a__U11#(tt()) -> c_2(a__U12#(tt())):9 -->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_2 mark#(tt()) -> c_16():16 -->_2 mark#(nil()) -> c_15():15 -->_1 a__U11#(X) -> c_1():8 -->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 5:S:mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)) -->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(tt()) -> c_16():16 -->_2 mark#(nil()) -> c_15():15 -->_1 a__U12#(tt()) -> c_4():11 -->_1 a__U12#(X) -> c_3():10 -->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 6:S:mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_3 mark#(tt()) -> c_16():16 -->_2 mark#(tt()) -> c_16():16 -->_3 mark#(nil()) -> c_15():15 -->_2 mark#(nil()) -> c_15():15 -->_1 a____#(X1,X2) -> c_6():12 -->_3 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_3 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_3 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 -->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 -->_1 a____#(nil(),X) -> c_8(mark#(X)):3 -->_1 a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)):2 -->_1 a____#(X,nil()) -> c_5(mark#(X)):1 7:S:mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)) -->_1 a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt())):14 -->_2 mark#(tt()) -> c_16():16 -->_2 mark#(nil()) -> c_15():15 -->_1 a__isNePal#(X) -> c_9():13 -->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 8:W:a__U11#(X) -> c_1() 9:W:a__U11#(tt()) -> c_2(a__U12#(tt())) -->_1 a__U12#(tt()) -> c_4():11 -->_1 a__U12#(X) -> c_3():10 10:W:a__U12#(X) -> c_3() 11:W:a__U12#(tt()) -> c_4() 12:W:a____#(X1,X2) -> c_6() 13:W:a__isNePal#(X) -> c_9() 14:W:a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt())) -->_1 a__U11#(tt()) -> c_2(a__U12#(tt())):9 -->_1 a__U11#(X) -> c_1():8 15:W:mark#(nil()) -> c_15() 16:W:mark#(tt()) -> c_16() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 12: a____#(X1,X2) -> c_6() 13: a__isNePal#(X) -> c_9() 15: mark#(nil()) -> c_15() 16: mark#(tt()) -> c_16() 14: a__isNePal#(__(I,__(P,I))) -> c_10(a__U11#(tt())) 8: a__U11#(X) -> c_1() 9: a__U11#(tt()) -> c_2(a__U12#(tt())) 10: a__U12#(X) -> c_3() 11: a__U12#(tt()) -> c_4() * Step 6: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: a____#(X,nil()) -> c_5(mark#(X)) a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)) a____#(nil(),X) -> c_8(mark#(X)) mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)) mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)) mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> a__U12(tt()) a__U12(X) -> U12(X) a__U12(tt()) -> tt() a____(X,nil()) -> mark(X) a____(X1,X2) -> __(X1,X2) a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(nil(),X) -> mark(X) a__isNePal(X) -> isNePal(X) a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() - Signature: {a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1 ,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/2,c_12/2 ,c_13/3,c_14/2,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal# ,mark#} and constructors {U11,U12,__,isNePal,nil,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:a____#(X,nil()) -> c_5(mark#(X)) -->_1 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_1 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_1 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_1 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 2:S:a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)) -->_5 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_4 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_5 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_4 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_5 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_4 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_5 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 -->_4 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 -->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 -->_3 a____#(nil(),X) -> c_8(mark#(X)):3 -->_1 a____#(nil(),X) -> c_8(mark#(X)):3 -->_3 a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)):2 -->_1 a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)):2 -->_3 a____#(X,nil()) -> c_5(mark#(X)):1 -->_1 a____#(X,nil()) -> c_5(mark#(X)):1 3:S:a____#(nil(),X) -> c_8(mark#(X)) -->_1 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_1 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_1 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_1 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 4:S:mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)) -->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 5:S:mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)) -->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 6:S:mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_3 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_3 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_3 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 -->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 -->_1 a____#(nil(),X) -> c_8(mark#(X)):3 -->_1 a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)):2 -->_1 a____#(X,nil()) -> c_5(mark#(X)):1 7:S:mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)) -->_2 mark#(isNePal(X)) -> c_14(a__isNePal#(mark(X)),mark#(X)):7 -->_2 mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(U12(X)) -> c_12(a__U12#(mark(X)),mark#(X)):5 -->_2 mark#(U11(X)) -> c_11(a__U11#(mark(X)),mark#(X)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mark#(U11(X)) -> c_11(mark#(X)) mark#(U12(X)) -> c_12(mark#(X)) mark#(isNePal(X)) -> c_14(mark#(X)) * Step 7: WeightGap MAYBE + Considered Problem: - Strict DPs: a____#(X,nil()) -> c_5(mark#(X)) a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)) a____#(nil(),X) -> c_8(mark#(X)) mark#(U11(X)) -> c_11(mark#(X)) mark#(U12(X)) -> c_12(mark#(X)) mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(isNePal(X)) -> c_14(mark#(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> a__U12(tt()) a__U12(X) -> U12(X) a__U12(tt()) -> tt() a____(X,nil()) -> mark(X) a____(X1,X2) -> __(X1,X2) a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(nil(),X) -> mark(X) a__isNePal(X) -> isNePal(X) a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() - Signature: {a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1 ,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/1,c_12/1 ,c_13/3,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal# ,mark#} and constructors {U11,U12,__,isNePal,nil,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, 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 (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(a__U11) = {1}, uargs(a__U12) = {1}, uargs(a____) = {1,2}, uargs(a__isNePal) = {1}, uargs(a____#) = {1,2}, uargs(c_5) = {1}, uargs(c_7) = {1,2,3,4,5}, uargs(c_8) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1,2,3}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(U11) = [0] p(U12) = [0] p(__) = [0] p(a__U11) = [1] x1 + [0] p(a__U12) = [1] x1 + [0] p(a____) = [1] x1 + [1] x2 + [0] p(a__isNePal) = [1] x1 + [0] p(isNePal) = [0] p(mark) = [0] p(nil) = [0] p(tt) = [0] p(a__U11#) = [0] p(a__U12#) = [0] p(a____#) = [1] x1 + [1] x2 + [4] p(a__isNePal#) = [0] p(mark#) = [2] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] x1 + [1] x2 + [1] x3 + [2] p(c_14) = [1] x1 + [4] p(c_15) = [0] p(c_16) = [1] Following rules are strictly oriented: a____#(X,nil()) = [1] X + [4] > [2] = c_5(mark#(X)) a____#(nil(),X) = [1] X + [4] > [2] = c_8(mark#(X)) Following rules are (at-least) weakly oriented: a____#(__(X,Y),Z) = [1] Z + [4] >= [14] = c_7(a____#(mark(X),a____(mark(Y),mark(Z))),mark#(X),a____#(mark(Y),mark(Z)),mark#(Y),mark#(Z)) mark#(U11(X)) = [2] >= [2] = c_11(mark#(X)) mark#(U12(X)) = [2] >= [2] = c_12(mark#(X)) mark#(__(X1,X2)) = [2] >= [10] = c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(isNePal(X)) = [2] >= [6] = c_14(mark#(X)) a__U11(X) = [1] X + [0] >= [0] = U11(X) a__U11(tt()) = [0] >= [0] = a__U12(tt()) a__U12(X) = [1] X + [0] >= [0] = U12(X) a__U12(tt()) = [0] >= [0] = tt() a____(X,nil()) = [1] X + [0] >= [0] = mark(X) a____(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = __(X1,X2) a____(__(X,Y),Z) = [1] Z + [0] >= [0] = a____(mark(X),a____(mark(Y),mark(Z))) a____(nil(),X) = [1] X + [0] >= [0] = mark(X) a__isNePal(X) = [1] X + [0] >= [0] = isNePal(X) a__isNePal(__(I,__(P,I))) = [0] >= [0] = a__U11(tt()) mark(U11(X)) = [0] >= [0] = a__U11(mark(X)) mark(U12(X)) = [0] >= [0] = a__U12(mark(X)) mark(__(X1,X2)) = [0] >= [0] = a____(mark(X1),mark(X2)) mark(isNePal(X)) = [0] >= [0] = a__isNePal(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(tt()) = [0] >= [0] = tt() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: Failure MAYBE + Considered Problem: - Strict DPs: a____#(__(X,Y),Z) -> c_7(a____#(mark(X),a____(mark(Y),mark(Z))) ,mark#(X) ,a____#(mark(Y),mark(Z)) ,mark#(Y) ,mark#(Z)) mark#(U11(X)) -> c_11(mark#(X)) mark#(U12(X)) -> c_12(mark#(X)) mark#(__(X1,X2)) -> c_13(a____#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(isNePal(X)) -> c_14(mark#(X)) - Weak DPs: a____#(X,nil()) -> c_5(mark#(X)) a____#(nil(),X) -> c_8(mark#(X)) - Weak TRS: a__U11(X) -> U11(X) a__U11(tt()) -> a__U12(tt()) a__U12(X) -> U12(X) a__U12(tt()) -> tt() a____(X,nil()) -> mark(X) a____(X1,X2) -> __(X1,X2) a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(nil(),X) -> mark(X) a__isNePal(X) -> isNePal(X) a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() - Signature: {a__U11/1,a__U12/1,a____/2,a__isNePal/1,mark/1,a__U11#/1,a__U12#/1,a____#/2,a__isNePal#/1,mark#/1} / {U11/1 ,U12/1,__/2,isNePal/1,nil/0,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/5,c_8/1,c_9/0,c_10/1,c_11/1,c_12/1 ,c_13/3,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a____#,a__isNePal# ,mark#} and constructors {U11,U12,__,isNePal,nil,tt} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE