MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(X) -> U12(X) a__U12(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X1,X2) -> U31(X1,X2) a__U31(tt(),N) -> mark(N) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) a__plus(X1,X2) -> plus(X1,X2) mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(isNat(X)) -> a__isNat(X) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() - Signature: {a__U11/2,a__U12/1,a__U21/1,a__U31/2,a__U41/3,a__U42/3,a__isNat/1,a__plus/2,mark/1} / {0/0,U11/2,U12/1,U21/1 ,U31/2,U41/3,U42/3,isNat/1,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11,a__U12,a__U21,a__U31,a__U41,a__U42,a__isNat ,a__plus,mark} and constructors {0,U11,U12,U21,U31,U41,U42,isNat,plus,s,tt} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__U11#(X1,X2) -> c_1() a__U11#(tt(),V2) -> c_2(a__U12#(a__isNat(V2)),a__isNat#(V2)) a__U12#(X) -> c_3() a__U12#(tt()) -> c_4() a__U21#(X) -> c_5() a__U21#(tt()) -> c_6() a__U31#(X1,X2) -> c_7() a__U31#(tt(),N) -> c_8(mark#(N)) a__U41#(X1,X2,X3) -> c_9() a__U41#(tt(),M,N) -> c_10(a__U42#(a__isNat(N),M,N),a__isNat#(N)) a__U42#(X1,X2,X3) -> c_11() a__U42#(tt(),M,N) -> c_12(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) a__isNat#(X) -> c_13() a__isNat#(0()) -> c_14() a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)) a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)) a__plus#(N,0()) -> c_17(a__U31#(a__isNat(N),N),a__isNat#(N)) a__plus#(N,s(M)) -> c_18(a__U41#(a__isNat(M),M,N),a__isNat#(M)) a__plus#(X1,X2) -> c_19() mark#(0()) -> c_20() mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)) mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)) mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)) mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)) mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)) mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)) mark#(isNat(X)) -> c_27(a__isNat#(X)) mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(s(X)) -> c_29(mark#(X)) mark#(tt()) -> c_30() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: a__U11#(X1,X2) -> c_1() a__U11#(tt(),V2) -> c_2(a__U12#(a__isNat(V2)),a__isNat#(V2)) a__U12#(X) -> c_3() a__U12#(tt()) -> c_4() a__U21#(X) -> c_5() a__U21#(tt()) -> c_6() a__U31#(X1,X2) -> c_7() a__U31#(tt(),N) -> c_8(mark#(N)) a__U41#(X1,X2,X3) -> c_9() a__U41#(tt(),M,N) -> c_10(a__U42#(a__isNat(N),M,N),a__isNat#(N)) a__U42#(X1,X2,X3) -> c_11() a__U42#(tt(),M,N) -> c_12(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) a__isNat#(X) -> c_13() a__isNat#(0()) -> c_14() a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)) a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)) a__plus#(N,0()) -> c_17(a__U31#(a__isNat(N),N),a__isNat#(N)) a__plus#(N,s(M)) -> c_18(a__U41#(a__isNat(M),M,N),a__isNat#(M)) a__plus#(X1,X2) -> c_19() mark#(0()) -> c_20() mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)) mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)) mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)) mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)) mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)) mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)) mark#(isNat(X)) -> c_27(a__isNat#(X)) mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(s(X)) -> c_29(mark#(X)) mark#(tt()) -> c_30() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(X) -> U12(X) a__U12(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X1,X2) -> U31(X1,X2) a__U31(tt(),N) -> mark(N) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) a__plus(X1,X2) -> plus(X1,X2) mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(isNat(X)) -> a__isNat(X) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() - Signature: {a__U11/2,a__U12/1,a__U21/1,a__U31/2,a__U41/3,a__U42/3,a__isNat/1,a__plus/2,mark/1,a__U11#/2,a__U12#/1 ,a__U21#/1,a__U31#/2,a__U41#/3,a__U42#/3,a__isNat#/1,a__plus#/2,mark#/1} / {0/0,U11/2,U12/1,U21/1,U31/2 ,U41/3,U42/3,isNat/1,plus/2,s/1,tt/0,c_1/0,c_2/2,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0 ,c_12/3,c_13/0,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/0,c_20/0,c_21/2,c_22/2,c_23/2,c_24/2,c_25/2,c_26/2 ,c_27/1,c_28/3,c_29/1,c_30/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a__U21#,a__U31#,a__U41#,a__U42#,a__isNat# ,a__plus#,mark#} and constructors {0,U11,U12,U21,U31,U41,U42,isNat,plus,s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,4,5,6,7,9,11,13,14,19,20,30} by application of Pre({1,3,4,5,6,7,9,11,13,14,19,20,30}) = {2,8,10,12,15,16,17,18,21,22,23,24,25,26,27,28,29}. Here rules are labelled as follows: 1: a__U11#(X1,X2) -> c_1() 2: a__U11#(tt(),V2) -> c_2(a__U12#(a__isNat(V2)),a__isNat#(V2)) 3: a__U12#(X) -> c_3() 4: a__U12#(tt()) -> c_4() 5: a__U21#(X) -> c_5() 6: a__U21#(tt()) -> c_6() 7: a__U31#(X1,X2) -> c_7() 8: a__U31#(tt(),N) -> c_8(mark#(N)) 9: a__U41#(X1,X2,X3) -> c_9() 10: a__U41#(tt(),M,N) -> c_10(a__U42#(a__isNat(N),M,N),a__isNat#(N)) 11: a__U42#(X1,X2,X3) -> c_11() 12: a__U42#(tt(),M,N) -> c_12(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) 13: a__isNat#(X) -> c_13() 14: a__isNat#(0()) -> c_14() 15: a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)) 16: a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)) 17: a__plus#(N,0()) -> c_17(a__U31#(a__isNat(N),N),a__isNat#(N)) 18: a__plus#(N,s(M)) -> c_18(a__U41#(a__isNat(M),M,N),a__isNat#(M)) 19: a__plus#(X1,X2) -> c_19() 20: mark#(0()) -> c_20() 21: mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)) 22: mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)) 23: mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)) 24: mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)) 25: mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)) 26: mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)) 27: mark#(isNat(X)) -> c_27(a__isNat#(X)) 28: mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) 29: mark#(s(X)) -> c_29(mark#(X)) 30: mark#(tt()) -> c_30() * Step 3: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: a__U11#(tt(),V2) -> c_2(a__U12#(a__isNat(V2)),a__isNat#(V2)) a__U31#(tt(),N) -> c_8(mark#(N)) a__U41#(tt(),M,N) -> c_10(a__U42#(a__isNat(N),M,N),a__isNat#(N)) a__U42#(tt(),M,N) -> c_12(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)) a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)) a__plus#(N,0()) -> c_17(a__U31#(a__isNat(N),N),a__isNat#(N)) a__plus#(N,s(M)) -> c_18(a__U41#(a__isNat(M),M,N),a__isNat#(M)) mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)) mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)) mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)) mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)) mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)) mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)) mark#(isNat(X)) -> c_27(a__isNat#(X)) mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(s(X)) -> c_29(mark#(X)) - Weak DPs: a__U11#(X1,X2) -> c_1() a__U12#(X) -> c_3() a__U12#(tt()) -> c_4() a__U21#(X) -> c_5() a__U21#(tt()) -> c_6() a__U31#(X1,X2) -> c_7() a__U41#(X1,X2,X3) -> c_9() a__U42#(X1,X2,X3) -> c_11() a__isNat#(X) -> c_13() a__isNat#(0()) -> c_14() a__plus#(X1,X2) -> c_19() mark#(0()) -> c_20() mark#(tt()) -> c_30() - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(X) -> U12(X) a__U12(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X1,X2) -> U31(X1,X2) a__U31(tt(),N) -> mark(N) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) a__plus(X1,X2) -> plus(X1,X2) mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(isNat(X)) -> a__isNat(X) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() - Signature: {a__U11/2,a__U12/1,a__U21/1,a__U31/2,a__U41/3,a__U42/3,a__isNat/1,a__plus/2,mark/1,a__U11#/2,a__U12#/1 ,a__U21#/1,a__U31#/2,a__U41#/3,a__U42#/3,a__isNat#/1,a__plus#/2,mark#/1} / {0/0,U11/2,U12/1,U21/1,U31/2 ,U41/3,U42/3,isNat/1,plus/2,s/1,tt/0,c_1/0,c_2/2,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0 ,c_12/3,c_13/0,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/0,c_20/0,c_21/2,c_22/2,c_23/2,c_24/2,c_25/2,c_26/2 ,c_27/1,c_28/3,c_29/1,c_30/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a__U21#,a__U31#,a__U41#,a__U42#,a__isNat# ,a__plus#,mark#} and constructors {0,U11,U12,U21,U31,U41,U42,isNat,plus,s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a__U11#(tt(),V2) -> c_2(a__U12#(a__isNat(V2)),a__isNat#(V2)) -->_2 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_2 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 -->_2 a__isNat#(0()) -> c_14():27 -->_2 a__isNat#(X) -> c_13():26 -->_1 a__U12#(tt()) -> c_4():20 -->_1 a__U12#(X) -> c_3():19 2:S:a__U31#(tt(),N) -> c_8(mark#(N)) -->_1 mark#(s(X)) -> c_29(mark#(X)):17 -->_1 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_1 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_1 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_1 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_1 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_1 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_1 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_1 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_1 mark#(tt()) -> c_30():30 -->_1 mark#(0()) -> c_20():29 3:S:a__U41#(tt(),M,N) -> c_10(a__U42#(a__isNat(N),M,N),a__isNat#(N)) -->_2 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_2 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 -->_1 a__U42#(tt(),M,N) -> c_12(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)):4 -->_2 a__isNat#(0()) -> c_14():27 -->_2 a__isNat#(X) -> c_13():26 -->_1 a__U42#(X1,X2,X3) -> c_11():25 4:S:a__U42#(tt(),M,N) -> c_12(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) -->_3 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_3 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_3 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_3 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_3 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_3 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_3 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_3 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_3 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_1 a__plus#(N,s(M)) -> c_18(a__U41#(a__isNat(M),M,N),a__isNat#(M)):8 -->_1 a__plus#(N,0()) -> c_17(a__U31#(a__isNat(N),N),a__isNat#(N)):7 -->_3 mark#(tt()) -> c_30():30 -->_2 mark#(tt()) -> c_30():30 -->_3 mark#(0()) -> c_20():29 -->_2 mark#(0()) -> c_20():29 -->_1 a__plus#(X1,X2) -> c_19():28 5:S:a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)) -->_2 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_2 a__isNat#(0()) -> c_14():27 -->_2 a__isNat#(X) -> c_13():26 -->_1 a__U11#(X1,X2) -> c_1():18 -->_2 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 -->_1 a__U11#(tt(),V2) -> c_2(a__U12#(a__isNat(V2)),a__isNat#(V2)):1 6:S:a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)) -->_2 a__isNat#(0()) -> c_14():27 -->_2 a__isNat#(X) -> c_13():26 -->_1 a__U21#(tt()) -> c_6():22 -->_1 a__U21#(X) -> c_5():21 -->_2 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_2 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 7:S:a__plus#(N,0()) -> c_17(a__U31#(a__isNat(N),N),a__isNat#(N)) -->_2 a__isNat#(0()) -> c_14():27 -->_2 a__isNat#(X) -> c_13():26 -->_1 a__U31#(X1,X2) -> c_7():23 -->_2 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_2 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 -->_1 a__U31#(tt(),N) -> c_8(mark#(N)):2 8:S:a__plus#(N,s(M)) -> c_18(a__U41#(a__isNat(M),M,N),a__isNat#(M)) -->_2 a__isNat#(0()) -> c_14():27 -->_2 a__isNat#(X) -> c_13():26 -->_1 a__U41#(X1,X2,X3) -> c_9():24 -->_2 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_2 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 -->_1 a__U41#(tt(),M,N) -> c_10(a__U42#(a__isNat(N),M,N),a__isNat#(N)):3 9:S:mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)) -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(tt()) -> c_30():30 -->_2 mark#(0()) -> c_20():29 -->_1 a__U11#(X1,X2) -> c_1():18 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_1 a__U11#(tt(),V2) -> c_2(a__U12#(a__isNat(V2)),a__isNat#(V2)):1 10:S:mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)) -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(tt()) -> c_30():30 -->_2 mark#(0()) -> c_20():29 -->_1 a__U12#(tt()) -> c_4():20 -->_1 a__U12#(X) -> c_3():19 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 11:S:mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)) -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(tt()) -> c_30():30 -->_2 mark#(0()) -> c_20():29 -->_1 a__U21#(tt()) -> c_6():22 -->_1 a__U21#(X) -> c_5():21 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 12:S:mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)) -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(tt()) -> c_30():30 -->_2 mark#(0()) -> c_20():29 -->_1 a__U31#(X1,X2) -> c_7():23 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_1 a__U31#(tt(),N) -> c_8(mark#(N)):2 13:S:mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(tt()) -> c_30():30 -->_2 mark#(0()) -> c_20():29 -->_1 a__U41#(X1,X2,X3) -> c_9():24 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_1 a__U41#(tt(),M,N) -> c_10(a__U42#(a__isNat(N),M,N),a__isNat#(N)):3 14:S:mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(tt()) -> c_30():30 -->_2 mark#(0()) -> c_20():29 -->_1 a__U42#(X1,X2,X3) -> c_11():25 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_1 a__U42#(tt(),M,N) -> c_12(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)):4 15:S:mark#(isNat(X)) -> c_27(a__isNat#(X)) -->_1 a__isNat#(0()) -> c_14():27 -->_1 a__isNat#(X) -> c_13():26 -->_1 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_1 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 16:S:mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_3 mark#(tt()) -> c_30():30 -->_2 mark#(tt()) -> c_30():30 -->_3 mark#(0()) -> c_20():29 -->_2 mark#(0()) -> c_20():29 -->_1 a__plus#(X1,X2) -> c_19():28 -->_3 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_3 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_3 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_3 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_3 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_3 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_3 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_3 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_1 a__plus#(N,s(M)) -> c_18(a__U41#(a__isNat(M),M,N),a__isNat#(M)):8 -->_1 a__plus#(N,0()) -> c_17(a__U31#(a__isNat(N),N),a__isNat#(N)):7 17:S:mark#(s(X)) -> c_29(mark#(X)) -->_1 mark#(tt()) -> c_30():30 -->_1 mark#(0()) -> c_20():29 -->_1 mark#(s(X)) -> c_29(mark#(X)):17 -->_1 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_1 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_1 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_1 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_1 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_1 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_1 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_1 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 18:W:a__U11#(X1,X2) -> c_1() 19:W:a__U12#(X) -> c_3() 20:W:a__U12#(tt()) -> c_4() 21:W:a__U21#(X) -> c_5() 22:W:a__U21#(tt()) -> c_6() 23:W:a__U31#(X1,X2) -> c_7() 24:W:a__U41#(X1,X2,X3) -> c_9() 25:W:a__U42#(X1,X2,X3) -> c_11() 26:W:a__isNat#(X) -> c_13() 27:W:a__isNat#(0()) -> c_14() 28:W:a__plus#(X1,X2) -> c_19() 29:W:mark#(0()) -> c_20() 30:W:mark#(tt()) -> c_30() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 23: a__U31#(X1,X2) -> c_7() 24: a__U41#(X1,X2,X3) -> c_9() 25: a__U42#(X1,X2,X3) -> c_11() 28: a__plus#(X1,X2) -> c_19() 29: mark#(0()) -> c_20() 30: mark#(tt()) -> c_30() 19: a__U12#(X) -> c_3() 20: a__U12#(tt()) -> c_4() 18: a__U11#(X1,X2) -> c_1() 21: a__U21#(X) -> c_5() 22: a__U21#(tt()) -> c_6() 26: a__isNat#(X) -> c_13() 27: a__isNat#(0()) -> c_14() * Step 4: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: a__U11#(tt(),V2) -> c_2(a__U12#(a__isNat(V2)),a__isNat#(V2)) a__U31#(tt(),N) -> c_8(mark#(N)) a__U41#(tt(),M,N) -> c_10(a__U42#(a__isNat(N),M,N),a__isNat#(N)) a__U42#(tt(),M,N) -> c_12(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)) a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)) a__plus#(N,0()) -> c_17(a__U31#(a__isNat(N),N),a__isNat#(N)) a__plus#(N,s(M)) -> c_18(a__U41#(a__isNat(M),M,N),a__isNat#(M)) mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)) mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)) mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)) mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)) mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)) mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)) mark#(isNat(X)) -> c_27(a__isNat#(X)) mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(s(X)) -> c_29(mark#(X)) - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(X) -> U12(X) a__U12(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X1,X2) -> U31(X1,X2) a__U31(tt(),N) -> mark(N) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) a__plus(X1,X2) -> plus(X1,X2) mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(isNat(X)) -> a__isNat(X) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() - Signature: {a__U11/2,a__U12/1,a__U21/1,a__U31/2,a__U41/3,a__U42/3,a__isNat/1,a__plus/2,mark/1,a__U11#/2,a__U12#/1 ,a__U21#/1,a__U31#/2,a__U41#/3,a__U42#/3,a__isNat#/1,a__plus#/2,mark#/1} / {0/0,U11/2,U12/1,U21/1,U31/2 ,U41/3,U42/3,isNat/1,plus/2,s/1,tt/0,c_1/0,c_2/2,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0 ,c_12/3,c_13/0,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/0,c_20/0,c_21/2,c_22/2,c_23/2,c_24/2,c_25/2,c_26/2 ,c_27/1,c_28/3,c_29/1,c_30/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a__U21#,a__U31#,a__U41#,a__U42#,a__isNat# ,a__plus#,mark#} and constructors {0,U11,U12,U21,U31,U41,U42,isNat,plus,s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:a__U11#(tt(),V2) -> c_2(a__U12#(a__isNat(V2)),a__isNat#(V2)) -->_2 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_2 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 2:S:a__U31#(tt(),N) -> c_8(mark#(N)) -->_1 mark#(s(X)) -> c_29(mark#(X)):17 -->_1 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_1 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_1 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_1 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_1 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_1 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_1 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_1 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 3:S:a__U41#(tt(),M,N) -> c_10(a__U42#(a__isNat(N),M,N),a__isNat#(N)) -->_2 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_2 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 -->_1 a__U42#(tt(),M,N) -> c_12(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)):4 4:S:a__U42#(tt(),M,N) -> c_12(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) -->_3 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_3 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_3 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_3 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_3 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_3 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_3 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_3 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_3 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_1 a__plus#(N,s(M)) -> c_18(a__U41#(a__isNat(M),M,N),a__isNat#(M)):8 -->_1 a__plus#(N,0()) -> c_17(a__U31#(a__isNat(N),N),a__isNat#(N)):7 5:S:a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)) -->_2 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_2 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 -->_1 a__U11#(tt(),V2) -> c_2(a__U12#(a__isNat(V2)),a__isNat#(V2)):1 6:S:a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)) -->_2 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_2 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 7:S:a__plus#(N,0()) -> c_17(a__U31#(a__isNat(N),N),a__isNat#(N)) -->_2 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_2 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 -->_1 a__U31#(tt(),N) -> c_8(mark#(N)):2 8:S:a__plus#(N,s(M)) -> c_18(a__U41#(a__isNat(M),M,N),a__isNat#(M)) -->_2 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_2 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 -->_1 a__U41#(tt(),M,N) -> c_10(a__U42#(a__isNat(N),M,N),a__isNat#(N)):3 9:S:mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)) -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_1 a__U11#(tt(),V2) -> c_2(a__U12#(a__isNat(V2)),a__isNat#(V2)):1 10:S:mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)) -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 11:S:mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)) -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 12:S:mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)) -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_1 a__U31#(tt(),N) -> c_8(mark#(N)):2 13:S:mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_1 a__U41#(tt(),M,N) -> c_10(a__U42#(a__isNat(N),M,N),a__isNat#(N)):3 14:S:mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_1 a__U42#(tt(),M,N) -> c_12(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)):4 15:S:mark#(isNat(X)) -> c_27(a__isNat#(X)) -->_1 a__isNat#(s(V1)) -> c_16(a__U21#(a__isNat(V1)),a__isNat#(V1)):6 -->_1 a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)):5 16:S:mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(s(X)) -> c_29(mark#(X)):17 -->_2 mark#(s(X)) -> c_29(mark#(X)):17 -->_3 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_2 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_3 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_2 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_3 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_2 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_3 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_2 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_3 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_2 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_3 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_2 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_3 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_2 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_3 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_2 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 -->_1 a__plus#(N,s(M)) -> c_18(a__U41#(a__isNat(M),M,N),a__isNat#(M)):8 -->_1 a__plus#(N,0()) -> c_17(a__U31#(a__isNat(N),N),a__isNat#(N)):7 17:S:mark#(s(X)) -> c_29(mark#(X)) -->_1 mark#(s(X)) -> c_29(mark#(X)):17 -->_1 mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):16 -->_1 mark#(isNat(X)) -> c_27(a__isNat#(X)):15 -->_1 mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)):14 -->_1 mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)):13 -->_1 mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)):12 -->_1 mark#(U21(X)) -> c_23(a__U21#(mark(X)),mark#(X)):11 -->_1 mark#(U12(X)) -> c_22(a__U12#(mark(X)),mark#(X)):10 -->_1 mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)):9 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: a__U11#(tt(),V2) -> c_2(a__isNat#(V2)) a__isNat#(s(V1)) -> c_16(a__isNat#(V1)) mark#(U12(X)) -> c_22(mark#(X)) mark#(U21(X)) -> c_23(mark#(X)) * Step 5: Failure MAYBE + Considered Problem: - Strict DPs: a__U11#(tt(),V2) -> c_2(a__isNat#(V2)) a__U31#(tt(),N) -> c_8(mark#(N)) a__U41#(tt(),M,N) -> c_10(a__U42#(a__isNat(N),M,N),a__isNat#(N)) a__U42#(tt(),M,N) -> c_12(a__plus#(mark(N),mark(M)),mark#(N),mark#(M)) a__isNat#(plus(V1,V2)) -> c_15(a__U11#(a__isNat(V1),V2),a__isNat#(V1)) a__isNat#(s(V1)) -> c_16(a__isNat#(V1)) a__plus#(N,0()) -> c_17(a__U31#(a__isNat(N),N),a__isNat#(N)) a__plus#(N,s(M)) -> c_18(a__U41#(a__isNat(M),M,N),a__isNat#(M)) mark#(U11(X1,X2)) -> c_21(a__U11#(mark(X1),X2),mark#(X1)) mark#(U12(X)) -> c_22(mark#(X)) mark#(U21(X)) -> c_23(mark#(X)) mark#(U31(X1,X2)) -> c_24(a__U31#(mark(X1),X2),mark#(X1)) mark#(U41(X1,X2,X3)) -> c_25(a__U41#(mark(X1),X2,X3),mark#(X1)) mark#(U42(X1,X2,X3)) -> c_26(a__U42#(mark(X1),X2,X3),mark#(X1)) mark#(isNat(X)) -> c_27(a__isNat#(X)) mark#(plus(X1,X2)) -> c_28(a__plus#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(s(X)) -> c_29(mark#(X)) - Weak TRS: a__U11(X1,X2) -> U11(X1,X2) a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(X) -> U12(X) a__U12(tt()) -> tt() a__U21(X) -> U21(X) a__U21(tt()) -> tt() a__U31(X1,X2) -> U31(X1,X2) a__U31(tt(),N) -> mark(N) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(X) -> isNat(X) a__isNat(0()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,0()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) a__plus(X1,X2) -> plus(X1,X2) mark(0()) -> 0() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(isNat(X)) -> a__isNat(X) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(tt()) -> tt() - Signature: {a__U11/2,a__U12/1,a__U21/1,a__U31/2,a__U41/3,a__U42/3,a__isNat/1,a__plus/2,mark/1,a__U11#/2,a__U12#/1 ,a__U21#/1,a__U31#/2,a__U41#/3,a__U42#/3,a__isNat#/1,a__plus#/2,mark#/1} / {0/0,U11/2,U12/1,U21/1,U31/2 ,U41/3,U42/3,isNat/1,plus/2,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0 ,c_12/3,c_13/0,c_14/0,c_15/2,c_16/1,c_17/2,c_18/2,c_19/0,c_20/0,c_21/2,c_22/1,c_23/1,c_24/2,c_25/2,c_26/2 ,c_27/1,c_28/3,c_29/1,c_30/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__U11#,a__U12#,a__U21#,a__U31#,a__U41#,a__U42#,a__isNat# ,a__plus#,mark#} and constructors {0,U11,U12,U21,U31,U41,U42,isNat,plus,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE