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