MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__x(X1,X2) -> x(X1,X2) mark(0()) -> 0() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) - Signature: {a__and/2,a__plus/2,a__x/2,mark/1} / {0/0,and/2,plus/2,s/1,tt/0,x/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__and,a__plus,a__x,mark} and constructors {0,and,plus,s ,tt,x} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__and#(X1,X2) -> c_1() a__and#(tt(),X) -> c_2(mark#(X)) a__plus#(N,0()) -> c_3(mark#(N)) a__plus#(N,s(M)) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) a__plus#(X1,X2) -> c_5() a__x#(N,0()) -> c_6() a__x#(N,s(M)) -> c_7(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)) a__x#(X1,X2) -> c_8() mark#(0()) -> c_9() mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)) mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(s(X)) -> c_12(mark#(X)) mark#(tt()) -> c_13() mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: a__and#(X1,X2) -> c_1() a__and#(tt(),X) -> c_2(mark#(X)) a__plus#(N,0()) -> c_3(mark#(N)) a__plus#(N,s(M)) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) a__plus#(X1,X2) -> c_5() a__x#(N,0()) -> c_6() a__x#(N,s(M)) -> c_7(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)) a__x#(X1,X2) -> c_8() mark#(0()) -> c_9() mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)) mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(s(X)) -> c_12(mark#(X)) mark#(tt()) -> c_13() mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak TRS: a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__x(X1,X2) -> x(X1,X2) mark(0()) -> 0() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) - Signature: {a__and/2,a__plus/2,a__x/2,mark/1,a__and#/2,a__plus#/2,a__x#/2,mark#/1} / {0/0,and/2,plus/2,s/1,tt/0,x/2 ,c_1/0,c_2/1,c_3/1,c_4/3,c_5/0,c_6/0,c_7/5,c_8/0,c_9/0,c_10/2,c_11/3,c_12/1,c_13/0,c_14/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__and#,a__plus#,a__x#,mark#} and constructors {0,and ,plus,s,tt,x} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,5,6,8,9,13} by application of Pre({1,5,6,8,9,13}) = {2,3,4,7,10,11,12,14}. Here rules are labelled as follows: 1: a__and#(X1,X2) -> c_1() 2: a__and#(tt(),X) -> c_2(mark#(X)) 3: a__plus#(N,0()) -> c_3(mark#(N)) 4: a__plus#(N,s(M)) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) 5: a__plus#(X1,X2) -> c_5() 6: a__x#(N,0()) -> c_6() 7: a__x#(N,s(M)) -> c_7(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)) 8: a__x#(X1,X2) -> c_8() 9: mark#(0()) -> c_9() 10: mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)) 11: mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) 12: mark#(s(X)) -> c_12(mark#(X)) 13: mark#(tt()) -> c_13() 14: mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) * Step 3: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: a__and#(tt(),X) -> c_2(mark#(X)) a__plus#(N,0()) -> c_3(mark#(N)) a__plus#(N,s(M)) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) a__x#(N,s(M)) -> c_7(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)) mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)) mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(s(X)) -> c_12(mark#(X)) mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak DPs: a__and#(X1,X2) -> c_1() a__plus#(X1,X2) -> c_5() a__x#(N,0()) -> c_6() a__x#(X1,X2) -> c_8() mark#(0()) -> c_9() mark#(tt()) -> c_13() - Weak TRS: a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__x(X1,X2) -> x(X1,X2) mark(0()) -> 0() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) - Signature: {a__and/2,a__plus/2,a__x/2,mark/1,a__and#/2,a__plus#/2,a__x#/2,mark#/1} / {0/0,and/2,plus/2,s/1,tt/0,x/2 ,c_1/0,c_2/1,c_3/1,c_4/3,c_5/0,c_6/0,c_7/5,c_8/0,c_9/0,c_10/2,c_11/3,c_12/1,c_13/0,c_14/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__and#,a__plus#,a__x#,mark#} and constructors {0,and ,plus,s,tt,x} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a__and#(tt(),X) -> c_2(mark#(X)) -->_1 mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):8 -->_1 mark#(s(X)) -> c_12(mark#(X)):7 -->_1 mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_1 mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)):5 -->_1 mark#(tt()) -> c_13():14 -->_1 mark#(0()) -> c_9():13 2:S:a__plus#(N,0()) -> c_3(mark#(N)) -->_1 mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):8 -->_1 mark#(s(X)) -> c_12(mark#(X)):7 -->_1 mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_1 mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)):5 -->_1 mark#(tt()) -> c_13():14 -->_1 mark#(0()) -> c_9():13 3:S:a__plus#(N,s(M)) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) -->_3 mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):8 -->_2 mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):8 -->_3 mark#(s(X)) -> c_12(mark#(X)):7 -->_2 mark#(s(X)) -> c_12(mark#(X)):7 -->_3 mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_3 mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)):5 -->_3 mark#(tt()) -> c_13():14 -->_2 mark#(tt()) -> c_13():14 -->_3 mark#(0()) -> c_9():13 -->_2 mark#(0()) -> c_9():13 -->_1 a__plus#(X1,X2) -> c_5():10 -->_1 a__plus#(N,s(M)) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)):3 -->_1 a__plus#(N,0()) -> c_3(mark#(N)):2 4:S:a__x#(N,s(M)) -> c_7(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)) -->_5 mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):8 -->_4 mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):8 -->_3 mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):8 -->_5 mark#(s(X)) -> c_12(mark#(X)):7 -->_4 mark#(s(X)) -> c_12(mark#(X)):7 -->_3 mark#(s(X)) -> c_12(mark#(X)):7 -->_5 mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_4 mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_3 mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_5 mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)):5 -->_4 mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)):5 -->_3 mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)):5 -->_5 mark#(tt()) -> c_13():14 -->_4 mark#(tt()) -> c_13():14 -->_3 mark#(tt()) -> c_13():14 -->_5 mark#(0()) -> c_9():13 -->_4 mark#(0()) -> c_9():13 -->_3 mark#(0()) -> c_9():13 -->_2 a__x#(X1,X2) -> c_8():12 -->_2 a__x#(N,0()) -> c_6():11 -->_1 a__plus#(X1,X2) -> c_5():10 -->_2 a__x#(N,s(M)) -> c_7(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)):4 -->_1 a__plus#(N,s(M)) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)):3 -->_1 a__plus#(N,0()) -> c_3(mark#(N)):2 5:S:mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)) -->_2 mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):8 -->_2 mark#(s(X)) -> c_12(mark#(X)):7 -->_2 mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(tt()) -> c_13():14 -->_2 mark#(0()) -> c_9():13 -->_1 a__and#(X1,X2) -> c_1():9 -->_2 mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)):5 -->_1 a__and#(tt(),X) -> c_2(mark#(X)):1 6:S:mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):8 -->_2 mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):8 -->_3 mark#(s(X)) -> c_12(mark#(X)):7 -->_2 mark#(s(X)) -> c_12(mark#(X)):7 -->_3 mark#(tt()) -> c_13():14 -->_2 mark#(tt()) -> c_13():14 -->_3 mark#(0()) -> c_9():13 -->_2 mark#(0()) -> c_9():13 -->_1 a__plus#(X1,X2) -> c_5():10 -->_3 mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_3 mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)):5 -->_1 a__plus#(N,s(M)) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)):3 -->_1 a__plus#(N,0()) -> c_3(mark#(N)):2 7:S:mark#(s(X)) -> c_12(mark#(X)) -->_1 mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):8 -->_1 mark#(tt()) -> c_13():14 -->_1 mark#(0()) -> c_9():13 -->_1 mark#(s(X)) -> c_12(mark#(X)):7 -->_1 mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_1 mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)):5 8:S:mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(tt()) -> c_13():14 -->_2 mark#(tt()) -> c_13():14 -->_3 mark#(0()) -> c_9():13 -->_2 mark#(0()) -> c_9():13 -->_1 a__x#(X1,X2) -> c_8():12 -->_1 a__x#(N,0()) -> c_6():11 -->_3 mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):8 -->_2 mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):8 -->_3 mark#(s(X)) -> c_12(mark#(X)):7 -->_2 mark#(s(X)) -> c_12(mark#(X)):7 -->_3 mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_3 mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)):5 -->_1 a__x#(N,s(M)) -> c_7(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)):4 9:W:a__and#(X1,X2) -> c_1() 10:W:a__plus#(X1,X2) -> c_5() 11:W:a__x#(N,0()) -> c_6() 12:W:a__x#(X1,X2) -> c_8() 13:W:mark#(0()) -> c_9() 14: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. 9: a__and#(X1,X2) -> c_1() 10: a__plus#(X1,X2) -> c_5() 11: a__x#(N,0()) -> c_6() 12: a__x#(X1,X2) -> c_8() 13: mark#(0()) -> c_9() 14: mark#(tt()) -> c_13() * Step 4: Failure MAYBE + Considered Problem: - Strict DPs: a__and#(tt(),X) -> c_2(mark#(X)) a__plus#(N,0()) -> c_3(mark#(N)) a__plus#(N,s(M)) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) a__x#(N,s(M)) -> c_7(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)) mark#(and(X1,X2)) -> c_10(a__and#(mark(X1),X2),mark#(X1)) mark#(plus(X1,X2)) -> c_11(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(s(X)) -> c_12(mark#(X)) mark#(x(X1,X2)) -> c_14(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak TRS: a__and(X1,X2) -> and(X1,X2) a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__x(X1,X2) -> x(X1,X2) mark(0()) -> 0() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) - Signature: {a__and/2,a__plus/2,a__x/2,mark/1,a__and#/2,a__plus#/2,a__x#/2,mark#/1} / {0/0,and/2,plus/2,s/1,tt/0,x/2 ,c_1/0,c_2/1,c_3/1,c_4/3,c_5/0,c_6/0,c_7/5,c_8/0,c_9/0,c_10/2,c_11/3,c_12/1,c_13/0,c_14/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__and#,a__plus#,a__x#,mark#} and constructors {0,and ,plus,s,tt,x} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE