MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U21(tt(),M,N) -> a__U22(tt(),M,N) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__U22(tt(),M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__U21(tt(),M,N) a__x(X1,X2) -> x(X1,X2) mark(0()) -> 0() mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) 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__U11/3,a__U12/3,a__U21/3,a__U22/3,a__plus/2,a__x/2,mark/1} / {0/0,U11/3,U12/3,U21/3,U22/3,plus/2,s/1,tt/0 ,x/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U12,a__U21,a__U22,a__plus,a__x ,mark} and constructors {0,U11,U12,U21,U22,plus,s,tt,x} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__U11#(X1,X2,X3) -> c_1() a__U11#(tt(),M,N) -> c_2(a__U12#(tt(),M,N)) a__U12#(X1,X2,X3) -> c_3() a__U12#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) a__U21#(X1,X2,X3) -> c_5() a__U21#(tt(),M,N) -> c_6(a__U22#(tt(),M,N)) a__U22#(X1,X2,X3) -> c_7() a__U22#(tt(),M,N) -> c_8(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)) a__plus#(N,0()) -> c_9(mark#(N)) a__plus#(N,s(M)) -> c_10(a__U11#(tt(),M,N)) a__plus#(X1,X2) -> c_11() a__x#(N,0()) -> c_12() a__x#(N,s(M)) -> c_13(a__U21#(tt(),M,N)) a__x#(X1,X2) -> c_14() mark#(0()) -> c_15() mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)) mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)) mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)) mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)) mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(s(X)) -> c_21(mark#(X)) mark#(tt()) -> c_22() mark#(x(X1,X2)) -> c_23(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__U11#(X1,X2,X3) -> c_1() a__U11#(tt(),M,N) -> c_2(a__U12#(tt(),M,N)) a__U12#(X1,X2,X3) -> c_3() a__U12#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) a__U21#(X1,X2,X3) -> c_5() a__U21#(tt(),M,N) -> c_6(a__U22#(tt(),M,N)) a__U22#(X1,X2,X3) -> c_7() a__U22#(tt(),M,N) -> c_8(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)) a__plus#(N,0()) -> c_9(mark#(N)) a__plus#(N,s(M)) -> c_10(a__U11#(tt(),M,N)) a__plus#(X1,X2) -> c_11() a__x#(N,0()) -> c_12() a__x#(N,s(M)) -> c_13(a__U21#(tt(),M,N)) a__x#(X1,X2) -> c_14() mark#(0()) -> c_15() mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)) mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)) mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)) mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)) mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(s(X)) -> c_21(mark#(X)) mark#(tt()) -> c_22() mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak TRS: a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U21(tt(),M,N) -> a__U22(tt(),M,N) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__U22(tt(),M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__U21(tt(),M,N) a__x(X1,X2) -> x(X1,X2) mark(0()) -> 0() mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) 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__U11/3,a__U12/3,a__U21/3,a__U22/3,a__plus/2,a__x/2,mark/1,a__U11#/3,a__U12#/3,a__U21#/3,a__U22#/3 ,a__plus#/2,a__x#/2,mark#/1} / {0/0,U11/3,U12/3,U21/3,U22/3,plus/2,s/1,tt/0,x/2,c_1/0,c_2/1,c_3/0,c_4/3 ,c_5/0,c_6/1,c_7/0,c_8/5,c_9/1,c_10/1,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0,c_16/2,c_17/2,c_18/2,c_19/2,c_20/3 ,c_21/1,c_22/0,c_23/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a__U21#,a__U22#,a__plus#,a__x# ,mark#} and constructors {0,U11,U12,U21,U22,plus,s,tt,x} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,5,7,11,12,14,15,22} by application of Pre({1,3,5,7,11,12,14,15,22}) = {2,4,6,8,9,10,13,16,17,18,19,20,21,23}. Here rules are labelled as follows: 1: a__U11#(X1,X2,X3) -> c_1() 2: a__U11#(tt(),M,N) -> c_2(a__U12#(tt(),M,N)) 3: a__U12#(X1,X2,X3) -> c_3() 4: a__U12#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) 5: a__U21#(X1,X2,X3) -> c_5() 6: a__U21#(tt(),M,N) -> c_6(a__U22#(tt(),M,N)) 7: a__U22#(X1,X2,X3) -> c_7() 8: a__U22#(tt(),M,N) -> c_8(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)) 9: a__plus#(N,0()) -> c_9(mark#(N)) 10: a__plus#(N,s(M)) -> c_10(a__U11#(tt(),M,N)) 11: a__plus#(X1,X2) -> c_11() 12: a__x#(N,0()) -> c_12() 13: a__x#(N,s(M)) -> c_13(a__U21#(tt(),M,N)) 14: a__x#(X1,X2) -> c_14() 15: mark#(0()) -> c_15() 16: mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)) 17: mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)) 18: mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)) 19: mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)) 20: mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) 21: mark#(s(X)) -> c_21(mark#(X)) 22: mark#(tt()) -> c_22() 23: mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) * Step 3: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: a__U11#(tt(),M,N) -> c_2(a__U12#(tt(),M,N)) a__U12#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) a__U21#(tt(),M,N) -> c_6(a__U22#(tt(),M,N)) a__U22#(tt(),M,N) -> c_8(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)) a__plus#(N,0()) -> c_9(mark#(N)) a__plus#(N,s(M)) -> c_10(a__U11#(tt(),M,N)) a__x#(N,s(M)) -> c_13(a__U21#(tt(),M,N)) mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)) mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)) mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)) mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)) mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(s(X)) -> c_21(mark#(X)) mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak DPs: a__U11#(X1,X2,X3) -> c_1() a__U12#(X1,X2,X3) -> c_3() a__U21#(X1,X2,X3) -> c_5() a__U22#(X1,X2,X3) -> c_7() a__plus#(X1,X2) -> c_11() a__x#(N,0()) -> c_12() a__x#(X1,X2) -> c_14() mark#(0()) -> c_15() mark#(tt()) -> c_22() - Weak TRS: a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U21(tt(),M,N) -> a__U22(tt(),M,N) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__U22(tt(),M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__U21(tt(),M,N) a__x(X1,X2) -> x(X1,X2) mark(0()) -> 0() mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) 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__U11/3,a__U12/3,a__U21/3,a__U22/3,a__plus/2,a__x/2,mark/1,a__U11#/3,a__U12#/3,a__U21#/3,a__U22#/3 ,a__plus#/2,a__x#/2,mark#/1} / {0/0,U11/3,U12/3,U21/3,U22/3,plus/2,s/1,tt/0,x/2,c_1/0,c_2/1,c_3/0,c_4/3 ,c_5/0,c_6/1,c_7/0,c_8/5,c_9/1,c_10/1,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0,c_16/2,c_17/2,c_18/2,c_19/2,c_20/3 ,c_21/1,c_22/0,c_23/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a__U21#,a__U22#,a__plus#,a__x# ,mark#} and constructors {0,U11,U12,U21,U22,plus,s,tt,x} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a__U11#(tt(),M,N) -> c_2(a__U12#(tt(),M,N)) -->_1 a__U12#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)):2 -->_1 a__U12#(X1,X2,X3) -> c_3():16 2:S:a__U12#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) -->_3 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_2 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_3 mark#(s(X)) -> c_21(mark#(X)):13 -->_2 mark#(s(X)) -> c_21(mark#(X)):13 -->_3 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_3 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_2 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_3 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_2 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_3 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_2 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_3 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_1 a__plus#(N,s(M)) -> c_10(a__U11#(tt(),M,N)):6 -->_1 a__plus#(N,0()) -> c_9(mark#(N)):5 -->_3 mark#(tt()) -> c_22():23 -->_2 mark#(tt()) -> c_22():23 -->_3 mark#(0()) -> c_15():22 -->_2 mark#(0()) -> c_15():22 -->_1 a__plus#(X1,X2) -> c_11():19 3:S:a__U21#(tt(),M,N) -> c_6(a__U22#(tt(),M,N)) -->_1 a__U22#(tt(),M,N) -> c_8(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__U22#(X1,X2,X3) -> c_7():18 4:S:a__U22#(tt(),M,N) -> c_8(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_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_4 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_3 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_5 mark#(s(X)) -> c_21(mark#(X)):13 -->_4 mark#(s(X)) -> c_21(mark#(X)):13 -->_3 mark#(s(X)) -> c_21(mark#(X)):13 -->_5 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_4 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_3 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_5 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_4 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_3 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_5 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_4 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_3 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_5 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_4 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_3 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_5 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_4 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_3 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_2 a__x#(N,s(M)) -> c_13(a__U21#(tt(),M,N)):7 -->_1 a__plus#(N,s(M)) -> c_10(a__U11#(tt(),M,N)):6 -->_1 a__plus#(N,0()) -> c_9(mark#(N)):5 -->_5 mark#(tt()) -> c_22():23 -->_4 mark#(tt()) -> c_22():23 -->_3 mark#(tt()) -> c_22():23 -->_5 mark#(0()) -> c_15():22 -->_4 mark#(0()) -> c_15():22 -->_3 mark#(0()) -> c_15():22 -->_2 a__x#(X1,X2) -> c_14():21 -->_2 a__x#(N,0()) -> c_12():20 -->_1 a__plus#(X1,X2) -> c_11():19 5:S:a__plus#(N,0()) -> c_9(mark#(N)) -->_1 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_1 mark#(s(X)) -> c_21(mark#(X)):13 -->_1 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_1 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_1 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_1 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(tt()) -> c_22():23 -->_1 mark#(0()) -> c_15():22 6:S:a__plus#(N,s(M)) -> c_10(a__U11#(tt(),M,N)) -->_1 a__U11#(X1,X2,X3) -> c_1():15 -->_1 a__U11#(tt(),M,N) -> c_2(a__U12#(tt(),M,N)):1 7:S:a__x#(N,s(M)) -> c_13(a__U21#(tt(),M,N)) -->_1 a__U21#(X1,X2,X3) -> c_5():17 -->_1 a__U21#(tt(),M,N) -> c_6(a__U22#(tt(),M,N)):3 8:S:mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_2 mark#(s(X)) -> c_21(mark#(X)):13 -->_2 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_2 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_2 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_2 mark#(tt()) -> c_22():23 -->_2 mark#(0()) -> c_15():22 -->_1 a__U11#(X1,X2,X3) -> c_1():15 -->_2 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_1 a__U11#(tt(),M,N) -> c_2(a__U12#(tt(),M,N)):1 9:S:mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_2 mark#(s(X)) -> c_21(mark#(X)):13 -->_2 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_2 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_2 mark#(tt()) -> c_22():23 -->_2 mark#(0()) -> c_15():22 -->_1 a__U12#(X1,X2,X3) -> c_3():16 -->_2 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_2 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_1 a__U12#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)):2 10:S:mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_2 mark#(s(X)) -> c_21(mark#(X)):13 -->_2 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_2 mark#(tt()) -> c_22():23 -->_2 mark#(0()) -> c_15():22 -->_1 a__U21#(X1,X2,X3) -> c_5():17 -->_2 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_2 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_2 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_1 a__U21#(tt(),M,N) -> c_6(a__U22#(tt(),M,N)):3 11:S:mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_2 mark#(s(X)) -> c_21(mark#(X)):13 -->_2 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(tt()) -> c_22():23 -->_2 mark#(0()) -> c_15():22 -->_1 a__U22#(X1,X2,X3) -> c_7():18 -->_2 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_2 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_2 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_2 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_1 a__U22#(tt(),M,N) -> c_8(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)):4 12:S:mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_2 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_3 mark#(s(X)) -> c_21(mark#(X)):13 -->_2 mark#(s(X)) -> c_21(mark#(X)):13 -->_3 mark#(tt()) -> c_22():23 -->_2 mark#(tt()) -> c_22():23 -->_3 mark#(0()) -> c_15():22 -->_2 mark#(0()) -> c_15():22 -->_1 a__plus#(X1,X2) -> c_11():19 -->_3 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_3 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_2 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_3 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_2 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_3 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_2 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_3 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_1 a__plus#(N,s(M)) -> c_10(a__U11#(tt(),M,N)):6 -->_1 a__plus#(N,0()) -> c_9(mark#(N)):5 13:S:mark#(s(X)) -> c_21(mark#(X)) -->_1 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_1 mark#(tt()) -> c_22():23 -->_1 mark#(0()) -> c_15():22 -->_1 mark#(s(X)) -> c_21(mark#(X)):13 -->_1 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_1 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_1 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_1 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 14:S:mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(tt()) -> c_22():23 -->_2 mark#(tt()) -> c_22():23 -->_3 mark#(0()) -> c_15():22 -->_2 mark#(0()) -> c_15():22 -->_1 a__x#(X1,X2) -> c_14():21 -->_1 a__x#(N,0()) -> c_12():20 -->_3 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_2 mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):14 -->_3 mark#(s(X)) -> c_21(mark#(X)):13 -->_2 mark#(s(X)) -> c_21(mark#(X)):13 -->_3 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_3 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_2 mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)):11 -->_3 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_2 mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)):10 -->_3 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_2 mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)):9 -->_3 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)):8 -->_1 a__x#(N,s(M)) -> c_13(a__U21#(tt(),M,N)):7 15:W:a__U11#(X1,X2,X3) -> c_1() 16:W:a__U12#(X1,X2,X3) -> c_3() 17:W:a__U21#(X1,X2,X3) -> c_5() 18:W:a__U22#(X1,X2,X3) -> c_7() 19:W:a__plus#(X1,X2) -> c_11() 20:W:a__x#(N,0()) -> c_12() 21:W:a__x#(X1,X2) -> c_14() 22:W:mark#(0()) -> c_15() 23:W:mark#(tt()) -> c_22() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 15: a__U11#(X1,X2,X3) -> c_1() 16: a__U12#(X1,X2,X3) -> c_3() 17: a__U21#(X1,X2,X3) -> c_5() 18: a__U22#(X1,X2,X3) -> c_7() 19: a__plus#(X1,X2) -> c_11() 20: a__x#(N,0()) -> c_12() 21: a__x#(X1,X2) -> c_14() 22: mark#(0()) -> c_15() 23: mark#(tt()) -> c_22() * Step 4: Failure MAYBE + Considered Problem: - Strict DPs: a__U11#(tt(),M,N) -> c_2(a__U12#(tt(),M,N)) a__U12#(tt(),M,N) -> c_4(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) a__U21#(tt(),M,N) -> c_6(a__U22#(tt(),M,N)) a__U22#(tt(),M,N) -> c_8(a__plus#(a__x(mark(N),mark(M)),mark(N)) ,a__x#(mark(N),mark(M)) ,mark#(N) ,mark#(M) ,mark#(N)) a__plus#(N,0()) -> c_9(mark#(N)) a__plus#(N,s(M)) -> c_10(a__U11#(tt(),M,N)) a__x#(N,s(M)) -> c_13(a__U21#(tt(),M,N)) mark#(U11(X1,X2,X3)) -> c_16(a__U11#(mark(X1),X2,X3),mark#(X1)) mark#(U12(X1,X2,X3)) -> c_17(a__U12#(mark(X1),X2,X3),mark#(X1)) mark#(U21(X1,X2,X3)) -> c_18(a__U21#(mark(X1),X2,X3),mark#(X1)) mark#(U22(X1,X2,X3)) -> c_19(a__U22#(mark(X1),X2,X3),mark#(X1)) mark#(plus(X1,X2)) -> c_20(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(s(X)) -> c_21(mark#(X)) mark#(x(X1,X2)) -> c_23(a__x#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak TRS: a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__U21(X1,X2,X3) -> U21(X1,X2,X3) a__U21(tt(),M,N) -> a__U22(tt(),M,N) a__U22(X1,X2,X3) -> U22(X1,X2,X3) a__U22(tt(),M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) a__plus(X1,X2) -> plus(X1,X2) a__x(N,0()) -> 0() a__x(N,s(M)) -> a__U21(tt(),M,N) a__x(X1,X2) -> x(X1,X2) mark(0()) -> 0() mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3) mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3) 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__U11/3,a__U12/3,a__U21/3,a__U22/3,a__plus/2,a__x/2,mark/1,a__U11#/3,a__U12#/3,a__U21#/3,a__U22#/3 ,a__plus#/2,a__x#/2,mark#/1} / {0/0,U11/3,U12/3,U21/3,U22/3,plus/2,s/1,tt/0,x/2,c_1/0,c_2/1,c_3/0,c_4/3 ,c_5/0,c_6/1,c_7/0,c_8/5,c_9/1,c_10/1,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0,c_16/2,c_17/2,c_18/2,c_19/2,c_20/3 ,c_21/1,c_22/0,c_23/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a__U21#,a__U22#,a__plus#,a__x# ,mark#} and constructors {0,U11,U12,U21,U22,plus,s,tt,x} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE