MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(a__add(mark(X),mark(Y))) a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__half(X) -> half(X) a__half(0()) -> 0() a__half(dbl(X)) -> mark(X) a__half(s(0())) -> 0() a__half(s(s(X))) -> s(a__half(mark(X))) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)),a__dbl(mark(X)))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(half(X)) -> a__half(mark(X)) mark(nil()) -> nil() mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Signature: {a__add/2,a__dbl/1,a__first/2,a__half/1,a__sqr/1,a__terms/1,mark/1} / {0/0,add/2,cons/2,dbl/1,first/2,half/1 ,nil/0,recip/1,s/1,sqr/1,terms/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__dbl,a__first,a__half,a__sqr,a__terms ,mark} and constructors {0,add,cons,dbl,first,half,nil,recip,s,sqr,terms} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__add#(X1,X2) -> c_1() a__add#(0(),X) -> c_2(mark#(X)) a__add#(s(X),Y) -> c_3(a__add#(mark(X),mark(Y)),mark#(X),mark#(Y)) a__dbl#(X) -> c_4() a__dbl#(0()) -> c_5() a__dbl#(s(X)) -> c_6(a__dbl#(mark(X)),mark#(X)) a__first#(X1,X2) -> c_7() a__first#(0(),X) -> c_8() a__first#(s(X),cons(Y,Z)) -> c_9(mark#(Y)) a__half#(X) -> c_10() a__half#(0()) -> c_11() a__half#(dbl(X)) -> c_12(mark#(X)) a__half#(s(0())) -> c_13() a__half#(s(s(X))) -> c_14(a__half#(mark(X)),mark#(X)) a__sqr#(X) -> c_15() a__sqr#(0()) -> c_16() a__sqr#(s(X)) -> c_17(a__add#(a__sqr(mark(X)),a__dbl(mark(X))) ,a__sqr#(mark(X)) ,mark#(X) ,a__dbl#(mark(X)) ,mark#(X)) a__terms#(N) -> c_18(a__sqr#(mark(N)),mark#(N)) a__terms#(X) -> c_19() mark#(0()) -> c_20() mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(cons(X1,X2)) -> c_22(mark#(X1)) mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)) mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)) mark#(nil()) -> c_26() mark#(recip(X)) -> c_27(mark#(X)) mark#(s(X)) -> c_28(mark#(X)) mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)) mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: a__add#(X1,X2) -> c_1() a__add#(0(),X) -> c_2(mark#(X)) a__add#(s(X),Y) -> c_3(a__add#(mark(X),mark(Y)),mark#(X),mark#(Y)) a__dbl#(X) -> c_4() a__dbl#(0()) -> c_5() a__dbl#(s(X)) -> c_6(a__dbl#(mark(X)),mark#(X)) a__first#(X1,X2) -> c_7() a__first#(0(),X) -> c_8() a__first#(s(X),cons(Y,Z)) -> c_9(mark#(Y)) a__half#(X) -> c_10() a__half#(0()) -> c_11() a__half#(dbl(X)) -> c_12(mark#(X)) a__half#(s(0())) -> c_13() a__half#(s(s(X))) -> c_14(a__half#(mark(X)),mark#(X)) a__sqr#(X) -> c_15() a__sqr#(0()) -> c_16() a__sqr#(s(X)) -> c_17(a__add#(a__sqr(mark(X)),a__dbl(mark(X))) ,a__sqr#(mark(X)) ,mark#(X) ,a__dbl#(mark(X)) ,mark#(X)) a__terms#(N) -> c_18(a__sqr#(mark(N)),mark#(N)) a__terms#(X) -> c_19() mark#(0()) -> c_20() mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(cons(X1,X2)) -> c_22(mark#(X1)) mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)) mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)) mark#(nil()) -> c_26() mark#(recip(X)) -> c_27(mark#(X)) mark#(s(X)) -> c_28(mark#(X)) mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)) mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(a__add(mark(X),mark(Y))) a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__half(X) -> half(X) a__half(0()) -> 0() a__half(dbl(X)) -> mark(X) a__half(s(0())) -> 0() a__half(s(s(X))) -> s(a__half(mark(X))) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)),a__dbl(mark(X)))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(half(X)) -> a__half(mark(X)) mark(nil()) -> nil() mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Signature: {a__add/2,a__dbl/1,a__first/2,a__half/1,a__sqr/1,a__terms/1,mark/1,a__add#/2,a__dbl#/1,a__first#/2 ,a__half#/1,a__sqr#/1,a__terms#/1,mark#/1} / {0/0,add/2,cons/2,dbl/1,first/2,half/1,nil/0,recip/1,s/1,sqr/1 ,terms/1,c_1/0,c_2/1,c_3/3,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/1,c_13/0,c_14/2,c_15/0 ,c_16/0,c_17/5,c_18/2,c_19/0,c_20/0,c_21/3,c_22/1,c_23/2,c_24/3,c_25/2,c_26/0,c_27/1,c_28/1,c_29/2,c_30/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__add#,a__dbl#,a__first#,a__half#,a__sqr#,a__terms# ,mark#} and constructors {0,add,cons,dbl,first,half,nil,recip,s,sqr,terms} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,5,7,8,10,11,13,15,16,19,20,26} by application of Pre({1,4,5,7,8,10,11,13,15,16,19,20,26}) = {2,3,6,9,12,14,17,18,21,22,23,24,25,27,28,29,30}. Here rules are labelled as follows: 1: a__add#(X1,X2) -> c_1() 2: a__add#(0(),X) -> c_2(mark#(X)) 3: a__add#(s(X),Y) -> c_3(a__add#(mark(X),mark(Y)),mark#(X),mark#(Y)) 4: a__dbl#(X) -> c_4() 5: a__dbl#(0()) -> c_5() 6: a__dbl#(s(X)) -> c_6(a__dbl#(mark(X)),mark#(X)) 7: a__first#(X1,X2) -> c_7() 8: a__first#(0(),X) -> c_8() 9: a__first#(s(X),cons(Y,Z)) -> c_9(mark#(Y)) 10: a__half#(X) -> c_10() 11: a__half#(0()) -> c_11() 12: a__half#(dbl(X)) -> c_12(mark#(X)) 13: a__half#(s(0())) -> c_13() 14: a__half#(s(s(X))) -> c_14(a__half#(mark(X)),mark#(X)) 15: a__sqr#(X) -> c_15() 16: a__sqr#(0()) -> c_16() 17: a__sqr#(s(X)) -> c_17(a__add#(a__sqr(mark(X)),a__dbl(mark(X))) ,a__sqr#(mark(X)) ,mark#(X) ,a__dbl#(mark(X)) ,mark#(X)) 18: a__terms#(N) -> c_18(a__sqr#(mark(N)),mark#(N)) 19: a__terms#(X) -> c_19() 20: mark#(0()) -> c_20() 21: mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) 22: mark#(cons(X1,X2)) -> c_22(mark#(X1)) 23: mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)) 24: mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) 25: mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)) 26: mark#(nil()) -> c_26() 27: mark#(recip(X)) -> c_27(mark#(X)) 28: mark#(s(X)) -> c_28(mark#(X)) 29: mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)) 30: mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)) * Step 3: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: a__add#(0(),X) -> c_2(mark#(X)) a__add#(s(X),Y) -> c_3(a__add#(mark(X),mark(Y)),mark#(X),mark#(Y)) a__dbl#(s(X)) -> c_6(a__dbl#(mark(X)),mark#(X)) a__first#(s(X),cons(Y,Z)) -> c_9(mark#(Y)) a__half#(dbl(X)) -> c_12(mark#(X)) a__half#(s(s(X))) -> c_14(a__half#(mark(X)),mark#(X)) a__sqr#(s(X)) -> c_17(a__add#(a__sqr(mark(X)),a__dbl(mark(X))) ,a__sqr#(mark(X)) ,mark#(X) ,a__dbl#(mark(X)) ,mark#(X)) a__terms#(N) -> c_18(a__sqr#(mark(N)),mark#(N)) mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(cons(X1,X2)) -> c_22(mark#(X1)) mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)) mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)) mark#(recip(X)) -> c_27(mark#(X)) mark#(s(X)) -> c_28(mark#(X)) mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)) mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)) - Weak DPs: a__add#(X1,X2) -> c_1() a__dbl#(X) -> c_4() a__dbl#(0()) -> c_5() a__first#(X1,X2) -> c_7() a__first#(0(),X) -> c_8() a__half#(X) -> c_10() a__half#(0()) -> c_11() a__half#(s(0())) -> c_13() a__sqr#(X) -> c_15() a__sqr#(0()) -> c_16() a__terms#(X) -> c_19() mark#(0()) -> c_20() mark#(nil()) -> c_26() - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(a__add(mark(X),mark(Y))) a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__half(X) -> half(X) a__half(0()) -> 0() a__half(dbl(X)) -> mark(X) a__half(s(0())) -> 0() a__half(s(s(X))) -> s(a__half(mark(X))) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)),a__dbl(mark(X)))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(half(X)) -> a__half(mark(X)) mark(nil()) -> nil() mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Signature: {a__add/2,a__dbl/1,a__first/2,a__half/1,a__sqr/1,a__terms/1,mark/1,a__add#/2,a__dbl#/1,a__first#/2 ,a__half#/1,a__sqr#/1,a__terms#/1,mark#/1} / {0/0,add/2,cons/2,dbl/1,first/2,half/1,nil/0,recip/1,s/1,sqr/1 ,terms/1,c_1/0,c_2/1,c_3/3,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/1,c_13/0,c_14/2,c_15/0 ,c_16/0,c_17/5,c_18/2,c_19/0,c_20/0,c_21/3,c_22/1,c_23/2,c_24/3,c_25/2,c_26/0,c_27/1,c_28/1,c_29/2,c_30/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__add#,a__dbl#,a__first#,a__half#,a__sqr#,a__terms# ,mark#} and constructors {0,add,cons,dbl,first,half,nil,recip,s,sqr,terms} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a__add#(0(),X) -> c_2(mark#(X)) -->_1 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_1 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_1 mark#(s(X)) -> c_28(mark#(X)):15 -->_1 mark#(recip(X)) -> c_27(mark#(X)):14 -->_1 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_1 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_1 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_1 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_1 mark#(nil()) -> c_26():30 -->_1 mark#(0()) -> c_20():29 2:S:a__add#(s(X),Y) -> c_3(a__add#(mark(X),mark(Y)),mark#(X),mark#(Y)) -->_3 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_2 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_3 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_2 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_3 mark#(s(X)) -> c_28(mark#(X)):15 -->_2 mark#(s(X)) -> c_28(mark#(X)):15 -->_3 mark#(recip(X)) -> c_27(mark#(X)):14 -->_2 mark#(recip(X)) -> c_27(mark#(X)):14 -->_3 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_2 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_3 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_3 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_2 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_3 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_2 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_3 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_2 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_3 mark#(nil()) -> c_26():30 -->_2 mark#(nil()) -> c_26():30 -->_3 mark#(0()) -> c_20():29 -->_2 mark#(0()) -> c_20():29 -->_1 a__add#(X1,X2) -> c_1():18 -->_1 a__add#(s(X),Y) -> c_3(a__add#(mark(X),mark(Y)),mark#(X),mark#(Y)):2 -->_1 a__add#(0(),X) -> c_2(mark#(X)):1 3:S:a__dbl#(s(X)) -> c_6(a__dbl#(mark(X)),mark#(X)) -->_2 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_2 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_2 mark#(s(X)) -> c_28(mark#(X)):15 -->_2 mark#(recip(X)) -> c_27(mark#(X)):14 -->_2 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_2 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_2 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_2 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_2 mark#(nil()) -> c_26():30 -->_2 mark#(0()) -> c_20():29 -->_1 a__dbl#(0()) -> c_5():20 -->_1 a__dbl#(X) -> c_4():19 -->_1 a__dbl#(s(X)) -> c_6(a__dbl#(mark(X)),mark#(X)):3 4:S:a__first#(s(X),cons(Y,Z)) -> c_9(mark#(Y)) -->_1 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_1 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_1 mark#(s(X)) -> c_28(mark#(X)):15 -->_1 mark#(recip(X)) -> c_27(mark#(X)):14 -->_1 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_1 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_1 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_1 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_1 mark#(nil()) -> c_26():30 -->_1 mark#(0()) -> c_20():29 5:S:a__half#(dbl(X)) -> c_12(mark#(X)) -->_1 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_1 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_1 mark#(s(X)) -> c_28(mark#(X)):15 -->_1 mark#(recip(X)) -> c_27(mark#(X)):14 -->_1 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_1 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_1 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_1 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_1 mark#(nil()) -> c_26():30 -->_1 mark#(0()) -> c_20():29 6:S:a__half#(s(s(X))) -> c_14(a__half#(mark(X)),mark#(X)) -->_2 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_2 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_2 mark#(s(X)) -> c_28(mark#(X)):15 -->_2 mark#(recip(X)) -> c_27(mark#(X)):14 -->_2 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_2 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_2 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_2 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_2 mark#(nil()) -> c_26():30 -->_2 mark#(0()) -> c_20():29 -->_1 a__half#(s(0())) -> c_13():25 -->_1 a__half#(0()) -> c_11():24 -->_1 a__half#(X) -> c_10():23 -->_1 a__half#(s(s(X))) -> c_14(a__half#(mark(X)),mark#(X)):6 -->_1 a__half#(dbl(X)) -> c_12(mark#(X)):5 7:S:a__sqr#(s(X)) -> c_17(a__add#(a__sqr(mark(X)),a__dbl(mark(X))) ,a__sqr#(mark(X)) ,mark#(X) ,a__dbl#(mark(X)) ,mark#(X)) -->_5 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_3 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_5 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_3 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_5 mark#(s(X)) -> c_28(mark#(X)):15 -->_3 mark#(s(X)) -> c_28(mark#(X)):15 -->_5 mark#(recip(X)) -> c_27(mark#(X)):14 -->_3 mark#(recip(X)) -> c_27(mark#(X)):14 -->_5 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_3 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_5 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_3 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_5 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_3 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_5 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_3 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_5 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_3 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_5 mark#(nil()) -> c_26():30 -->_3 mark#(nil()) -> c_26():30 -->_5 mark#(0()) -> c_20():29 -->_3 mark#(0()) -> c_20():29 -->_2 a__sqr#(0()) -> c_16():27 -->_2 a__sqr#(X) -> c_15():26 -->_4 a__dbl#(0()) -> c_5():20 -->_4 a__dbl#(X) -> c_4():19 -->_1 a__add#(X1,X2) -> c_1():18 -->_2 a__sqr#(s(X)) -> c_17(a__add#(a__sqr(mark(X)),a__dbl(mark(X))) ,a__sqr#(mark(X)) ,mark#(X) ,a__dbl#(mark(X)) ,mark#(X)):7 -->_4 a__dbl#(s(X)) -> c_6(a__dbl#(mark(X)),mark#(X)):3 -->_1 a__add#(s(X),Y) -> c_3(a__add#(mark(X),mark(Y)),mark#(X),mark#(Y)):2 -->_1 a__add#(0(),X) -> c_2(mark#(X)):1 8:S:a__terms#(N) -> c_18(a__sqr#(mark(N)),mark#(N)) -->_2 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_2 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_2 mark#(s(X)) -> c_28(mark#(X)):15 -->_2 mark#(recip(X)) -> c_27(mark#(X)):14 -->_2 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_2 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_2 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_2 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_2 mark#(nil()) -> c_26():30 -->_2 mark#(0()) -> c_20():29 -->_1 a__sqr#(0()) -> c_16():27 -->_1 a__sqr#(X) -> c_15():26 -->_1 a__sqr#(s(X)) -> c_17(a__add#(a__sqr(mark(X)),a__dbl(mark(X))) ,a__sqr#(mark(X)) ,mark#(X) ,a__dbl#(mark(X)) ,mark#(X)):7 9:S:mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_2 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_3 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_2 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_3 mark#(s(X)) -> c_28(mark#(X)):15 -->_2 mark#(s(X)) -> c_28(mark#(X)):15 -->_3 mark#(recip(X)) -> c_27(mark#(X)):14 -->_2 mark#(recip(X)) -> c_27(mark#(X)):14 -->_3 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_2 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_3 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_3 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_2 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_3 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_2 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_3 mark#(nil()) -> c_26():30 -->_2 mark#(nil()) -> c_26():30 -->_3 mark#(0()) -> c_20():29 -->_2 mark#(0()) -> c_20():29 -->_1 a__add#(X1,X2) -> c_1():18 -->_3 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_2 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_1 a__add#(s(X),Y) -> c_3(a__add#(mark(X),mark(Y)),mark#(X),mark#(Y)):2 -->_1 a__add#(0(),X) -> c_2(mark#(X)):1 10:S:mark#(cons(X1,X2)) -> c_22(mark#(X1)) -->_1 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_1 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_1 mark#(s(X)) -> c_28(mark#(X)):15 -->_1 mark#(recip(X)) -> c_27(mark#(X)):14 -->_1 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_1 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_1 mark#(nil()) -> c_26():30 -->_1 mark#(0()) -> c_20():29 -->_1 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_1 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 11:S:mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)) -->_2 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_2 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_2 mark#(s(X)) -> c_28(mark#(X)):15 -->_2 mark#(recip(X)) -> c_27(mark#(X)):14 -->_2 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_2 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(nil()) -> c_26():30 -->_2 mark#(0()) -> c_20():29 -->_1 a__dbl#(0()) -> c_5():20 -->_1 a__dbl#(X) -> c_4():19 -->_2 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_2 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_2 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_1 a__dbl#(s(X)) -> c_6(a__dbl#(mark(X)),mark#(X)):3 12:S:mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_2 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_3 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_2 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_3 mark#(s(X)) -> c_28(mark#(X)):15 -->_2 mark#(s(X)) -> c_28(mark#(X)):15 -->_3 mark#(recip(X)) -> c_27(mark#(X)):14 -->_2 mark#(recip(X)) -> c_27(mark#(X)):14 -->_3 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_2 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_3 mark#(nil()) -> c_26():30 -->_2 mark#(nil()) -> c_26():30 -->_3 mark#(0()) -> c_20():29 -->_2 mark#(0()) -> c_20():29 -->_1 a__first#(0(),X) -> c_8():22 -->_1 a__first#(X1,X2) -> c_7():21 -->_3 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_3 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_2 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_3 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_2 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_3 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_2 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_1 a__first#(s(X),cons(Y,Z)) -> c_9(mark#(Y)):4 13:S:mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)) -->_2 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_2 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_2 mark#(s(X)) -> c_28(mark#(X)):15 -->_2 mark#(recip(X)) -> c_27(mark#(X)):14 -->_2 mark#(nil()) -> c_26():30 -->_2 mark#(0()) -> c_20():29 -->_1 a__half#(s(0())) -> c_13():25 -->_1 a__half#(0()) -> c_11():24 -->_1 a__half#(X) -> c_10():23 -->_2 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_2 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_2 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_2 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_1 a__half#(s(s(X))) -> c_14(a__half#(mark(X)),mark#(X)):6 -->_1 a__half#(dbl(X)) -> c_12(mark#(X)):5 14:S:mark#(recip(X)) -> c_27(mark#(X)) -->_1 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_1 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_1 mark#(s(X)) -> c_28(mark#(X)):15 -->_1 mark#(nil()) -> c_26():30 -->_1 mark#(0()) -> c_20():29 -->_1 mark#(recip(X)) -> c_27(mark#(X)):14 -->_1 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_1 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_1 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_1 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 15:S:mark#(s(X)) -> c_28(mark#(X)) -->_1 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_1 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_1 mark#(nil()) -> c_26():30 -->_1 mark#(0()) -> c_20():29 -->_1 mark#(s(X)) -> c_28(mark#(X)):15 -->_1 mark#(recip(X)) -> c_27(mark#(X)):14 -->_1 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_1 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_1 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_1 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 16:S:mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)) -->_2 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_2 mark#(nil()) -> c_26():30 -->_2 mark#(0()) -> c_20():29 -->_1 a__sqr#(0()) -> c_16():27 -->_1 a__sqr#(X) -> c_15():26 -->_2 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_2 mark#(s(X)) -> c_28(mark#(X)):15 -->_2 mark#(recip(X)) -> c_27(mark#(X)):14 -->_2 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_2 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_2 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_2 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_1 a__sqr#(s(X)) -> c_17(a__add#(a__sqr(mark(X)),a__dbl(mark(X))) ,a__sqr#(mark(X)) ,mark#(X) ,a__dbl#(mark(X)) ,mark#(X)):7 17:S:mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)) -->_2 mark#(nil()) -> c_26():30 -->_2 mark#(0()) -> c_20():29 -->_1 a__terms#(X) -> c_19():28 -->_2 mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)):17 -->_2 mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)):16 -->_2 mark#(s(X)) -> c_28(mark#(X)):15 -->_2 mark#(recip(X)) -> c_27(mark#(X)):14 -->_2 mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)):13 -->_2 mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)):11 -->_2 mark#(cons(X1,X2)) -> c_22(mark#(X1)):10 -->_2 mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_1 a__terms#(N) -> c_18(a__sqr#(mark(N)),mark#(N)):8 18:W:a__add#(X1,X2) -> c_1() 19:W:a__dbl#(X) -> c_4() 20:W:a__dbl#(0()) -> c_5() 21:W:a__first#(X1,X2) -> c_7() 22:W:a__first#(0(),X) -> c_8() 23:W:a__half#(X) -> c_10() 24:W:a__half#(0()) -> c_11() 25:W:a__half#(s(0())) -> c_13() 26:W:a__sqr#(X) -> c_15() 27:W:a__sqr#(0()) -> c_16() 28:W:a__terms#(X) -> c_19() 29:W:mark#(0()) -> c_20() 30:W:mark#(nil()) -> c_26() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 18: a__add#(X1,X2) -> c_1() 19: a__dbl#(X) -> c_4() 20: a__dbl#(0()) -> c_5() 21: a__first#(X1,X2) -> c_7() 22: a__first#(0(),X) -> c_8() 23: a__half#(X) -> c_10() 24: a__half#(0()) -> c_11() 25: a__half#(s(0())) -> c_13() 26: a__sqr#(X) -> c_15() 27: a__sqr#(0()) -> c_16() 28: a__terms#(X) -> c_19() 29: mark#(0()) -> c_20() 30: mark#(nil()) -> c_26() * Step 4: Failure MAYBE + Considered Problem: - Strict DPs: a__add#(0(),X) -> c_2(mark#(X)) a__add#(s(X),Y) -> c_3(a__add#(mark(X),mark(Y)),mark#(X),mark#(Y)) a__dbl#(s(X)) -> c_6(a__dbl#(mark(X)),mark#(X)) a__first#(s(X),cons(Y,Z)) -> c_9(mark#(Y)) a__half#(dbl(X)) -> c_12(mark#(X)) a__half#(s(s(X))) -> c_14(a__half#(mark(X)),mark#(X)) a__sqr#(s(X)) -> c_17(a__add#(a__sqr(mark(X)),a__dbl(mark(X))) ,a__sqr#(mark(X)) ,mark#(X) ,a__dbl#(mark(X)) ,mark#(X)) a__terms#(N) -> c_18(a__sqr#(mark(N)),mark#(N)) mark#(add(X1,X2)) -> c_21(a__add#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(cons(X1,X2)) -> c_22(mark#(X1)) mark#(dbl(X)) -> c_23(a__dbl#(mark(X)),mark#(X)) mark#(first(X1,X2)) -> c_24(a__first#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(half(X)) -> c_25(a__half#(mark(X)),mark#(X)) mark#(recip(X)) -> c_27(mark#(X)) mark#(s(X)) -> c_28(mark#(X)) mark#(sqr(X)) -> c_29(a__sqr#(mark(X)),mark#(X)) mark#(terms(X)) -> c_30(a__terms#(mark(X)),mark#(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(a__add(mark(X),mark(Y))) a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z)) a__half(X) -> half(X) a__half(0()) -> 0() a__half(dbl(X)) -> mark(X) a__half(s(0())) -> 0() a__half(s(s(X))) -> s(a__half(mark(X))) a__sqr(X) -> sqr(X) a__sqr(0()) -> 0() a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)),a__dbl(mark(X)))) a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N))) a__terms(X) -> terms(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(half(X)) -> a__half(mark(X)) mark(nil()) -> nil() mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(terms(X)) -> a__terms(mark(X)) - Signature: {a__add/2,a__dbl/1,a__first/2,a__half/1,a__sqr/1,a__terms/1,mark/1,a__add#/2,a__dbl#/1,a__first#/2 ,a__half#/1,a__sqr#/1,a__terms#/1,mark#/1} / {0/0,add/2,cons/2,dbl/1,first/2,half/1,nil/0,recip/1,s/1,sqr/1 ,terms/1,c_1/0,c_2/1,c_3/3,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/1,c_13/0,c_14/2,c_15/0 ,c_16/0,c_17/5,c_18/2,c_19/0,c_20/0,c_21/3,c_22/1,c_23/2,c_24/3,c_25/2,c_26/0,c_27/1,c_28/1,c_29/2,c_30/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__add#,a__dbl#,a__first#,a__half#,a__sqr#,a__terms# ,mark#} and constructors {0,add,cons,dbl,first,half,nil,recip,s,sqr,terms} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE