MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(X) -> dbls(X) a__dbls(cons(X,Y)) -> cons(dbl(X),dbls(Y)) a__dbls(nil()) -> nil() a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__indx(X1,X2) -> indx(X1,X2) a__indx(cons(X,Y),Z) -> cons(sel(X,Z),indx(Y,Z)) a__indx(nil(),X) -> nil() a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,Y)) -> mark(X) a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(from(X)) -> a__from(X) mark(indx(X1,X2)) -> a__indx(mark(X1),X2) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) - Signature: {a__dbl/1,a__dbls/1,a__from/1,a__indx/2,a__sel/2,mark/1} / {0/0,cons/2,dbl/1,dbls/1,from/1,indx/2,nil/0,s/1 ,sel/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__dbl,a__dbls,a__from,a__indx,a__sel ,mark} and constructors {0,cons,dbl,dbls,from,indx,nil,s,sel} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__dbl#(X) -> c_1() a__dbl#(0()) -> c_2() a__dbl#(s(X)) -> c_3() a__dbls#(X) -> c_4() a__dbls#(cons(X,Y)) -> c_5() a__dbls#(nil()) -> c_6() a__from#(X) -> c_7() a__from#(X) -> c_8() a__indx#(X1,X2) -> c_9() a__indx#(cons(X,Y),Z) -> c_10() a__indx#(nil(),X) -> c_11() a__sel#(X1,X2) -> c_12() a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)) a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) mark#(0()) -> c_15() mark#(cons(X1,X2)) -> c_16() mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)) mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)) mark#(from(X)) -> c_19(a__from#(X)) mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)) mark#(nil()) -> c_21() mark#(s(X)) -> c_22() mark#(sel(X1,X2)) -> c_23(a__sel#(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__dbl#(X) -> c_1() a__dbl#(0()) -> c_2() a__dbl#(s(X)) -> c_3() a__dbls#(X) -> c_4() a__dbls#(cons(X,Y)) -> c_5() a__dbls#(nil()) -> c_6() a__from#(X) -> c_7() a__from#(X) -> c_8() a__indx#(X1,X2) -> c_9() a__indx#(cons(X,Y),Z) -> c_10() a__indx#(nil(),X) -> c_11() a__sel#(X1,X2) -> c_12() a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)) a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) mark#(0()) -> c_15() mark#(cons(X1,X2)) -> c_16() mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)) mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)) mark#(from(X)) -> c_19(a__from#(X)) mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)) mark#(nil()) -> c_21() mark#(s(X)) -> c_22() mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak TRS: a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(X) -> dbls(X) a__dbls(cons(X,Y)) -> cons(dbl(X),dbls(Y)) a__dbls(nil()) -> nil() a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__indx(X1,X2) -> indx(X1,X2) a__indx(cons(X,Y),Z) -> cons(sel(X,Z),indx(Y,Z)) a__indx(nil(),X) -> nil() a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,Y)) -> mark(X) a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(from(X)) -> a__from(X) mark(indx(X1,X2)) -> a__indx(mark(X1),X2) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) - Signature: {a__dbl/1,a__dbls/1,a__from/1,a__indx/2,a__sel/2,mark/1,a__dbl#/1,a__dbls#/1,a__from#/1,a__indx#/2,a__sel#/2 ,mark#/1} / {0/0,cons/2,dbl/1,dbls/1,from/1,indx/2,nil/0,s/1,sel/2,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/3,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/0,c_22/0 ,c_23/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__dbl#,a__dbls#,a__from#,a__indx#,a__sel# ,mark#} and constructors {0,cons,dbl,dbls,from,indx,nil,s,sel} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,5,6,7,8,9,10,11,12,15,16,21,22} by application of Pre({1,2,3,4,5,6,7,8,9,10,11,12,15,16,21,22}) = {13,14,17,18,19,20,23}. Here rules are labelled as follows: 1: a__dbl#(X) -> c_1() 2: a__dbl#(0()) -> c_2() 3: a__dbl#(s(X)) -> c_3() 4: a__dbls#(X) -> c_4() 5: a__dbls#(cons(X,Y)) -> c_5() 6: a__dbls#(nil()) -> c_6() 7: a__from#(X) -> c_7() 8: a__from#(X) -> c_8() 9: a__indx#(X1,X2) -> c_9() 10: a__indx#(cons(X,Y),Z) -> c_10() 11: a__indx#(nil(),X) -> c_11() 12: a__sel#(X1,X2) -> c_12() 13: a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)) 14: a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) 15: mark#(0()) -> c_15() 16: mark#(cons(X1,X2)) -> c_16() 17: mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)) 18: mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)) 19: mark#(from(X)) -> c_19(a__from#(X)) 20: mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)) 21: mark#(nil()) -> c_21() 22: mark#(s(X)) -> c_22() 23: mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)) a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)) mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)) mark#(from(X)) -> c_19(a__from#(X)) mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)) mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak DPs: a__dbl#(X) -> c_1() a__dbl#(0()) -> c_2() a__dbl#(s(X)) -> c_3() a__dbls#(X) -> c_4() a__dbls#(cons(X,Y)) -> c_5() a__dbls#(nil()) -> c_6() a__from#(X) -> c_7() a__from#(X) -> c_8() a__indx#(X1,X2) -> c_9() a__indx#(cons(X,Y),Z) -> c_10() a__indx#(nil(),X) -> c_11() a__sel#(X1,X2) -> c_12() mark#(0()) -> c_15() mark#(cons(X1,X2)) -> c_16() mark#(nil()) -> c_21() mark#(s(X)) -> c_22() - Weak TRS: a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(X) -> dbls(X) a__dbls(cons(X,Y)) -> cons(dbl(X),dbls(Y)) a__dbls(nil()) -> nil() a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__indx(X1,X2) -> indx(X1,X2) a__indx(cons(X,Y),Z) -> cons(sel(X,Z),indx(Y,Z)) a__indx(nil(),X) -> nil() a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,Y)) -> mark(X) a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(from(X)) -> a__from(X) mark(indx(X1,X2)) -> a__indx(mark(X1),X2) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) - Signature: {a__dbl/1,a__dbls/1,a__from/1,a__indx/2,a__sel/2,mark/1,a__dbl#/1,a__dbls#/1,a__from#/1,a__indx#/2,a__sel#/2 ,mark#/1} / {0/0,cons/2,dbl/1,dbls/1,from/1,indx/2,nil/0,s/1,sel/2,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/3,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/0,c_22/0 ,c_23/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__dbl#,a__dbls#,a__from#,a__indx#,a__sel# ,mark#} and constructors {0,cons,dbl,dbls,from,indx,nil,s,sel} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5} by application of Pre({5}) = {1,2,3,4,6,7}. Here rules are labelled as follows: 1: a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)) 2: a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) 3: mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)) 4: mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)) 5: mark#(from(X)) -> c_19(a__from#(X)) 6: mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)) 7: mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) 8: a__dbl#(X) -> c_1() 9: a__dbl#(0()) -> c_2() 10: a__dbl#(s(X)) -> c_3() 11: a__dbls#(X) -> c_4() 12: a__dbls#(cons(X,Y)) -> c_5() 13: a__dbls#(nil()) -> c_6() 14: a__from#(X) -> c_7() 15: a__from#(X) -> c_8() 16: a__indx#(X1,X2) -> c_9() 17: a__indx#(cons(X,Y),Z) -> c_10() 18: a__indx#(nil(),X) -> c_11() 19: a__sel#(X1,X2) -> c_12() 20: mark#(0()) -> c_15() 21: mark#(cons(X1,X2)) -> c_16() 22: mark#(nil()) -> c_21() 23: mark#(s(X)) -> c_22() * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)) a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)) mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)) mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)) mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak DPs: a__dbl#(X) -> c_1() a__dbl#(0()) -> c_2() a__dbl#(s(X)) -> c_3() a__dbls#(X) -> c_4() a__dbls#(cons(X,Y)) -> c_5() a__dbls#(nil()) -> c_6() a__from#(X) -> c_7() a__from#(X) -> c_8() a__indx#(X1,X2) -> c_9() a__indx#(cons(X,Y),Z) -> c_10() a__indx#(nil(),X) -> c_11() a__sel#(X1,X2) -> c_12() mark#(0()) -> c_15() mark#(cons(X1,X2)) -> c_16() mark#(from(X)) -> c_19(a__from#(X)) mark#(nil()) -> c_21() mark#(s(X)) -> c_22() - Weak TRS: a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(X) -> dbls(X) a__dbls(cons(X,Y)) -> cons(dbl(X),dbls(Y)) a__dbls(nil()) -> nil() a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__indx(X1,X2) -> indx(X1,X2) a__indx(cons(X,Y),Z) -> cons(sel(X,Z),indx(Y,Z)) a__indx(nil(),X) -> nil() a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,Y)) -> mark(X) a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(from(X)) -> a__from(X) mark(indx(X1,X2)) -> a__indx(mark(X1),X2) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) - Signature: {a__dbl/1,a__dbls/1,a__from/1,a__indx/2,a__sel/2,mark/1,a__dbl#/1,a__dbls#/1,a__from#/1,a__indx#/2,a__sel#/2 ,mark#/1} / {0/0,cons/2,dbl/1,dbls/1,from/1,indx/2,nil/0,s/1,sel/2,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/3,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/0,c_22/0 ,c_23/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__dbl#,a__dbls#,a__from#,a__indx#,a__sel# ,mark#} and constructors {0,cons,dbl,dbls,from,indx,nil,s,sel} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)) -->_1 mark#(from(X)) -> c_19(a__from#(X)):21 -->_1 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_1 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_1 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_1 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 -->_1 mark#(s(X)) -> c_22():23 -->_1 mark#(nil()) -> c_21():22 -->_1 mark#(cons(X1,X2)) -> c_16():20 -->_1 mark#(0()) -> c_15():19 2:S:a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) -->_3 mark#(from(X)) -> c_19(a__from#(X)):21 -->_2 mark#(from(X)) -> c_19(a__from#(X)):21 -->_3 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_3 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_3 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_2 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_3 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 -->_2 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 -->_3 mark#(s(X)) -> c_22():23 -->_2 mark#(s(X)) -> c_22():23 -->_3 mark#(nil()) -> c_21():22 -->_2 mark#(nil()) -> c_21():22 -->_3 mark#(cons(X1,X2)) -> c_16():20 -->_2 mark#(cons(X1,X2)) -> c_16():20 -->_3 mark#(0()) -> c_15():19 -->_2 mark#(0()) -> c_15():19 -->_1 a__sel#(X1,X2) -> c_12():18 -->_1 a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)):2 -->_1 a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)):1 3:S:mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)) -->_2 mark#(from(X)) -> c_19(a__from#(X)):21 -->_2 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_2 mark#(s(X)) -> c_22():23 -->_2 mark#(nil()) -> c_21():22 -->_2 mark#(cons(X1,X2)) -> c_16():20 -->_2 mark#(0()) -> c_15():19 -->_1 a__dbl#(s(X)) -> c_3():9 -->_1 a__dbl#(0()) -> c_2():8 -->_1 a__dbl#(X) -> c_1():7 -->_2 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 4:S:mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)) -->_2 mark#(from(X)) -> c_19(a__from#(X)):21 -->_2 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(s(X)) -> c_22():23 -->_2 mark#(nil()) -> c_21():22 -->_2 mark#(cons(X1,X2)) -> c_16():20 -->_2 mark#(0()) -> c_15():19 -->_1 a__dbls#(nil()) -> c_6():12 -->_1 a__dbls#(cons(X,Y)) -> c_5():11 -->_1 a__dbls#(X) -> c_4():10 -->_2 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_2 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 5:S:mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)) -->_2 mark#(from(X)) -> c_19(a__from#(X)):21 -->_2 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(s(X)) -> c_22():23 -->_2 mark#(nil()) -> c_21():22 -->_2 mark#(cons(X1,X2)) -> c_16():20 -->_2 mark#(0()) -> c_15():19 -->_1 a__indx#(nil(),X) -> c_11():17 -->_1 a__indx#(cons(X,Y),Z) -> c_10():16 -->_1 a__indx#(X1,X2) -> c_9():15 -->_2 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_2 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 6:S:mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(from(X)) -> c_19(a__from#(X)):21 -->_2 mark#(from(X)) -> c_19(a__from#(X)):21 -->_3 mark#(s(X)) -> c_22():23 -->_2 mark#(s(X)) -> c_22():23 -->_3 mark#(nil()) -> c_21():22 -->_2 mark#(nil()) -> c_21():22 -->_3 mark#(cons(X1,X2)) -> c_16():20 -->_2 mark#(cons(X1,X2)) -> c_16():20 -->_3 mark#(0()) -> c_15():19 -->_2 mark#(0()) -> c_15():19 -->_1 a__sel#(X1,X2) -> c_12():18 -->_3 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_3 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_3 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_2 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_3 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 -->_2 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 -->_1 a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)):2 -->_1 a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)):1 7:W:a__dbl#(X) -> c_1() 8:W:a__dbl#(0()) -> c_2() 9:W:a__dbl#(s(X)) -> c_3() 10:W:a__dbls#(X) -> c_4() 11:W:a__dbls#(cons(X,Y)) -> c_5() 12:W:a__dbls#(nil()) -> c_6() 13:W:a__from#(X) -> c_7() 14:W:a__from#(X) -> c_8() 15:W:a__indx#(X1,X2) -> c_9() 16:W:a__indx#(cons(X,Y),Z) -> c_10() 17:W:a__indx#(nil(),X) -> c_11() 18:W:a__sel#(X1,X2) -> c_12() 19:W:mark#(0()) -> c_15() 20:W:mark#(cons(X1,X2)) -> c_16() 21:W:mark#(from(X)) -> c_19(a__from#(X)) -->_1 a__from#(X) -> c_8():14 -->_1 a__from#(X) -> c_7():13 22:W:mark#(nil()) -> c_21() 23:W:mark#(s(X)) -> c_22() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: a__dbl#(X) -> c_1() 8: a__dbl#(0()) -> c_2() 9: a__dbl#(s(X)) -> c_3() 10: a__dbls#(X) -> c_4() 11: a__dbls#(cons(X,Y)) -> c_5() 12: a__dbls#(nil()) -> c_6() 15: a__indx#(X1,X2) -> c_9() 16: a__indx#(cons(X,Y),Z) -> c_10() 17: a__indx#(nil(),X) -> c_11() 18: a__sel#(X1,X2) -> c_12() 19: mark#(0()) -> c_15() 20: mark#(cons(X1,X2)) -> c_16() 22: mark#(nil()) -> c_21() 23: mark#(s(X)) -> c_22() 21: mark#(from(X)) -> c_19(a__from#(X)) 13: a__from#(X) -> c_7() 14: a__from#(X) -> c_8() * Step 5: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)) a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)) mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)) mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)) mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak TRS: a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(X) -> dbls(X) a__dbls(cons(X,Y)) -> cons(dbl(X),dbls(Y)) a__dbls(nil()) -> nil() a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__indx(X1,X2) -> indx(X1,X2) a__indx(cons(X,Y),Z) -> cons(sel(X,Z),indx(Y,Z)) a__indx(nil(),X) -> nil() a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,Y)) -> mark(X) a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(from(X)) -> a__from(X) mark(indx(X1,X2)) -> a__indx(mark(X1),X2) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) - Signature: {a__dbl/1,a__dbls/1,a__from/1,a__indx/2,a__sel/2,mark/1,a__dbl#/1,a__dbls#/1,a__from#/1,a__indx#/2,a__sel#/2 ,mark#/1} / {0/0,cons/2,dbl/1,dbls/1,from/1,indx/2,nil/0,s/1,sel/2,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/3,c_15/0,c_16/0,c_17/2,c_18/2,c_19/1,c_20/2,c_21/0,c_22/0 ,c_23/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__dbl#,a__dbls#,a__from#,a__indx#,a__sel# ,mark#} and constructors {0,cons,dbl,dbls,from,indx,nil,s,sel} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)) -->_1 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_1 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_1 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_1 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 2:S:a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) -->_3 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_3 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_3 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_2 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_3 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 -->_2 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 -->_1 a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)):2 -->_1 a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)):1 3:S:mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)) -->_2 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_2 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 4:S:mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)) -->_2 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_2 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 5:S:mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)) -->_2 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_2 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 6:S:mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_2 mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):6 -->_3 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_2 mark#(indx(X1,X2)) -> c_20(a__indx#(mark(X1),X2),mark#(X1)):5 -->_3 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_2 mark#(dbls(X)) -> c_18(a__dbls#(mark(X)),mark#(X)):4 -->_3 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 -->_2 mark#(dbl(X)) -> c_17(a__dbl#(mark(X)),mark#(X)):3 -->_1 a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)):2 -->_1 a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mark#(dbl(X)) -> c_17(mark#(X)) mark#(dbls(X)) -> c_18(mark#(X)) mark#(indx(X1,X2)) -> c_20(mark#(X1)) * Step 6: WeightGap MAYBE + Considered Problem: - Strict DPs: a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)) a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) mark#(dbl(X)) -> c_17(mark#(X)) mark#(dbls(X)) -> c_18(mark#(X)) mark#(indx(X1,X2)) -> c_20(mark#(X1)) mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak TRS: a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(X) -> dbls(X) a__dbls(cons(X,Y)) -> cons(dbl(X),dbls(Y)) a__dbls(nil()) -> nil() a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__indx(X1,X2) -> indx(X1,X2) a__indx(cons(X,Y),Z) -> cons(sel(X,Z),indx(Y,Z)) a__indx(nil(),X) -> nil() a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,Y)) -> mark(X) a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(from(X)) -> a__from(X) mark(indx(X1,X2)) -> a__indx(mark(X1),X2) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) - Signature: {a__dbl/1,a__dbls/1,a__from/1,a__indx/2,a__sel/2,mark/1,a__dbl#/1,a__dbls#/1,a__from#/1,a__indx#/2,a__sel#/2 ,mark#/1} / {0/0,cons/2,dbl/1,dbls/1,from/1,indx/2,nil/0,s/1,sel/2,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/3,c_15/0,c_16/0,c_17/1,c_18/1,c_19/1,c_20/1,c_21/0,c_22/0 ,c_23/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__dbl#,a__dbls#,a__from#,a__indx#,a__sel# ,mark#} and constructors {0,cons,dbl,dbls,from,indx,nil,s,sel} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__dbl) = {1}, uargs(a__dbls) = {1}, uargs(a__indx) = {1}, uargs(a__sel) = {1,2}, uargs(a__sel#) = {1,2}, uargs(c_13) = {1}, uargs(c_14) = {1,2,3}, uargs(c_17) = {1}, uargs(c_18) = {1}, uargs(c_20) = {1}, uargs(c_23) = {1,2,3} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__dbl) = [1] x1 + [0] p(a__dbls) = [1] x1 + [0] p(a__from) = [0] p(a__indx) = [1] x1 + [0] p(a__sel) = [1] x1 + [1] x2 + [0] p(cons) = [0] p(dbl) = [0] p(dbls) = [0] p(from) = [0] p(indx) = [0] p(mark) = [0] p(nil) = [0] p(s) = [0] p(sel) = [0] p(a__dbl#) = [2] x1 + [4] p(a__dbls#) = [1] p(a__from#) = [0] p(a__indx#) = [1] x1 + [4] x2 + [0] p(a__sel#) = [1] x1 + [1] x2 + [6] p(mark#) = [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [1] x2 + [1] x3 + [0] p(c_15) = [1] p(c_16) = [1] p(c_17) = [1] x1 + [0] p(c_18) = [1] x1 + [0] p(c_19) = [0] p(c_20) = [1] x1 + [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [1] x1 + [1] x2 + [1] x3 + [0] Following rules are strictly oriented: a__sel#(0(),cons(X,Y)) = [6] > [1] = c_13(mark#(X)) Following rules are (at-least) weakly oriented: a__sel#(s(X),cons(Y,Z)) = [6] >= [8] = c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) mark#(dbl(X)) = [1] >= [1] = c_17(mark#(X)) mark#(dbls(X)) = [1] >= [1] = c_18(mark#(X)) mark#(indx(X1,X2)) = [1] >= [1] = c_20(mark#(X1)) mark#(sel(X1,X2)) = [1] >= [8] = c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) a__dbl(X) = [1] X + [0] >= [0] = dbl(X) a__dbl(0()) = [0] >= [0] = 0() a__dbl(s(X)) = [0] >= [0] = s(s(dbl(X))) a__dbls(X) = [1] X + [0] >= [0] = dbls(X) a__dbls(cons(X,Y)) = [0] >= [0] = cons(dbl(X),dbls(Y)) a__dbls(nil()) = [0] >= [0] = nil() a__from(X) = [0] >= [0] = cons(X,from(s(X))) a__from(X) = [0] >= [0] = from(X) a__indx(X1,X2) = [1] X1 + [0] >= [0] = indx(X1,X2) a__indx(cons(X,Y),Z) = [0] >= [0] = cons(sel(X,Z),indx(Y,Z)) a__indx(nil(),X) = [0] >= [0] = nil() a__sel(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = sel(X1,X2) a__sel(0(),cons(X,Y)) = [0] >= [0] = mark(X) a__sel(s(X),cons(Y,Z)) = [0] >= [0] = a__sel(mark(X),mark(Z)) mark(0()) = [0] >= [0] = 0() mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(dbl(X)) = [0] >= [0] = a__dbl(mark(X)) mark(dbls(X)) = [0] >= [0] = a__dbls(mark(X)) mark(from(X)) = [0] >= [0] = a__from(X) mark(indx(X1,X2)) = [0] >= [0] = a__indx(mark(X1),X2) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) mark(sel(X1,X2)) = [0] >= [0] = a__sel(mark(X1),mark(X2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: Failure MAYBE + Considered Problem: - Strict DPs: a__sel#(s(X),cons(Y,Z)) -> c_14(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) mark#(dbl(X)) -> c_17(mark#(X)) mark#(dbls(X)) -> c_18(mark#(X)) mark#(indx(X1,X2)) -> c_20(mark#(X1)) mark#(sel(X1,X2)) -> c_23(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak DPs: a__sel#(0(),cons(X,Y)) -> c_13(mark#(X)) - Weak TRS: a__dbl(X) -> dbl(X) a__dbl(0()) -> 0() a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(X) -> dbls(X) a__dbls(cons(X,Y)) -> cons(dbl(X),dbls(Y)) a__dbls(nil()) -> nil() a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__indx(X1,X2) -> indx(X1,X2) a__indx(cons(X,Y),Z) -> cons(sel(X,Z),indx(Y,Z)) a__indx(nil(),X) -> nil() a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,Y)) -> mark(X) a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(from(X)) -> a__from(X) mark(indx(X1,X2)) -> a__indx(mark(X1),X2) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) - Signature: {a__dbl/1,a__dbls/1,a__from/1,a__indx/2,a__sel/2,mark/1,a__dbl#/1,a__dbls#/1,a__from#/1,a__indx#/2,a__sel#/2 ,mark#/1} / {0/0,cons/2,dbl/1,dbls/1,from/1,indx/2,nil/0,s/1,sel/2,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/3,c_15/0,c_16/0,c_17/1,c_18/1,c_19/1,c_20/1,c_21/0,c_22/0 ,c_23/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__dbl#,a__dbls#,a__from#,a__indx#,a__sel# ,mark#} and constructors {0,cons,dbl,dbls,from,indx,nil,s,sel} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE