WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__eq(X,Y) -> false() a__eq(X1,X2) -> eq(X1,X2) a__eq(0(),0()) -> true() a__eq(s(X),s(Y)) -> a__eq(X,Y) a__inf(X) -> cons(X,inf(s(X))) a__inf(X) -> inf(X) a__length(X) -> length(X) a__length(cons(X,L)) -> s(length(L)) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),X) -> nil() a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(eq(X1,X2)) -> a__eq(X1,X2) mark(false()) -> false() mark(inf(X)) -> a__inf(mark(X)) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(true()) -> true() - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1} / {0/0,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq,a__inf,a__length,a__take,mark} and constructors {0 ,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__eq#(X,Y) -> c_1() a__eq#(X1,X2) -> c_2() a__eq#(0(),0()) -> c_3() a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) a__inf#(X) -> c_5() a__inf#(X) -> c_6() a__length#(X) -> c_7() a__length#(cons(X,L)) -> c_8() a__length#(nil()) -> c_9() a__take#(X1,X2) -> c_10() a__take#(0(),X) -> c_11() a__take#(s(X),cons(Y,L)) -> c_12() mark#(0()) -> c_13() mark#(cons(X1,X2)) -> c_14() mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) mark#(false()) -> c_16() mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)) mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)) mark#(nil()) -> c_19() mark#(s(X)) -> c_20() mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(true()) -> c_22() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__eq#(X,Y) -> c_1() a__eq#(X1,X2) -> c_2() a__eq#(0(),0()) -> c_3() a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) a__inf#(X) -> c_5() a__inf#(X) -> c_6() a__length#(X) -> c_7() a__length#(cons(X,L)) -> c_8() a__length#(nil()) -> c_9() a__take#(X1,X2) -> c_10() a__take#(0(),X) -> c_11() a__take#(s(X),cons(Y,L)) -> c_12() mark#(0()) -> c_13() mark#(cons(X1,X2)) -> c_14() mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) mark#(false()) -> c_16() mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)) mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)) mark#(nil()) -> c_19() mark#(s(X)) -> c_20() mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(true()) -> c_22() - Weak TRS: a__eq(X,Y) -> false() a__eq(X1,X2) -> eq(X1,X2) a__eq(0(),0()) -> true() a__eq(s(X),s(Y)) -> a__eq(X,Y) a__inf(X) -> cons(X,inf(s(X))) a__inf(X) -> inf(X) a__length(X) -> length(X) a__length(cons(X,L)) -> s(length(L)) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),X) -> nil() a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(eq(X1,X2)) -> a__eq(X1,X2) mark(false()) -> false() mark(inf(X)) -> a__inf(mark(X)) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(true()) -> true() - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1,a__eq#/2,a__inf#/1,a__length#/1,a__take#/2,mark#/1} / {0/0 ,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0,c_1/0,c_2/0,c_3/0,c_4/1,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_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/2,c_19/0,c_20/0,c_21/3,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq#,a__inf#,a__length#,a__take# ,mark#} and constructors {0,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,5,6,7,8,9,10,11,12,13,14,16,19,20,22} by application of Pre({1,2,3,5,6,7,8,9,10,11,12,13,14,16,19,20,22}) = {4,15,17,18,21}. Here rules are labelled as follows: 1: a__eq#(X,Y) -> c_1() 2: a__eq#(X1,X2) -> c_2() 3: a__eq#(0(),0()) -> c_3() 4: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) 5: a__inf#(X) -> c_5() 6: a__inf#(X) -> c_6() 7: a__length#(X) -> c_7() 8: a__length#(cons(X,L)) -> c_8() 9: a__length#(nil()) -> c_9() 10: a__take#(X1,X2) -> c_10() 11: a__take#(0(),X) -> c_11() 12: a__take#(s(X),cons(Y,L)) -> c_12() 13: mark#(0()) -> c_13() 14: mark#(cons(X1,X2)) -> c_14() 15: mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) 16: mark#(false()) -> c_16() 17: mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)) 18: mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)) 19: mark#(nil()) -> c_19() 20: mark#(s(X)) -> c_20() 21: mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) 22: mark#(true()) -> c_22() * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)) mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)) mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak DPs: a__eq#(X,Y) -> c_1() a__eq#(X1,X2) -> c_2() a__eq#(0(),0()) -> c_3() a__inf#(X) -> c_5() a__inf#(X) -> c_6() a__length#(X) -> c_7() a__length#(cons(X,L)) -> c_8() a__length#(nil()) -> c_9() a__take#(X1,X2) -> c_10() a__take#(0(),X) -> c_11() a__take#(s(X),cons(Y,L)) -> c_12() mark#(0()) -> c_13() mark#(cons(X1,X2)) -> c_14() mark#(false()) -> c_16() mark#(nil()) -> c_19() mark#(s(X)) -> c_20() mark#(true()) -> c_22() - Weak TRS: a__eq(X,Y) -> false() a__eq(X1,X2) -> eq(X1,X2) a__eq(0(),0()) -> true() a__eq(s(X),s(Y)) -> a__eq(X,Y) a__inf(X) -> cons(X,inf(s(X))) a__inf(X) -> inf(X) a__length(X) -> length(X) a__length(cons(X,L)) -> s(length(L)) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),X) -> nil() a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(eq(X1,X2)) -> a__eq(X1,X2) mark(false()) -> false() mark(inf(X)) -> a__inf(mark(X)) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(true()) -> true() - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1,a__eq#/2,a__inf#/1,a__length#/1,a__take#/2,mark#/1} / {0/0 ,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0,c_1/0,c_2/0,c_3/0,c_4/1,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_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/2,c_19/0,c_20/0,c_21/3,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq#,a__inf#,a__length#,a__take# ,mark#} and constructors {0,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) -->_1 a__eq#(0(),0()) -> c_3():8 -->_1 a__eq#(X1,X2) -> c_2():7 -->_1 a__eq#(X,Y) -> c_1():6 -->_1 a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)):1 2:S:mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) -->_1 a__eq#(0(),0()) -> c_3():8 -->_1 a__eq#(X1,X2) -> c_2():7 -->_1 a__eq#(X,Y) -> c_1():6 -->_1 a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)):1 3:S:mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)) -->_2 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):5 -->_2 mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)):4 -->_2 mark#(true()) -> c_22():22 -->_2 mark#(s(X)) -> c_20():21 -->_2 mark#(nil()) -> c_19():20 -->_2 mark#(false()) -> c_16():19 -->_2 mark#(cons(X1,X2)) -> c_14():18 -->_2 mark#(0()) -> c_13():17 -->_1 a__inf#(X) -> c_6():10 -->_1 a__inf#(X) -> c_5():9 -->_2 mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)):3 -->_2 mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)):2 4:S:mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)) -->_2 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):5 -->_2 mark#(true()) -> c_22():22 -->_2 mark#(s(X)) -> c_20():21 -->_2 mark#(nil()) -> c_19():20 -->_2 mark#(false()) -> c_16():19 -->_2 mark#(cons(X1,X2)) -> c_14():18 -->_2 mark#(0()) -> c_13():17 -->_1 a__length#(nil()) -> c_9():13 -->_1 a__length#(cons(X,L)) -> c_8():12 -->_1 a__length#(X) -> c_7():11 -->_2 mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)):4 -->_2 mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)):3 -->_2 mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)):2 5:S:mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(true()) -> c_22():22 -->_2 mark#(true()) -> c_22():22 -->_3 mark#(s(X)) -> c_20():21 -->_2 mark#(s(X)) -> c_20():21 -->_3 mark#(nil()) -> c_19():20 -->_2 mark#(nil()) -> c_19():20 -->_3 mark#(false()) -> c_16():19 -->_2 mark#(false()) -> c_16():19 -->_3 mark#(cons(X1,X2)) -> c_14():18 -->_2 mark#(cons(X1,X2)) -> c_14():18 -->_3 mark#(0()) -> c_13():17 -->_2 mark#(0()) -> c_13():17 -->_1 a__take#(s(X),cons(Y,L)) -> c_12():16 -->_1 a__take#(0(),X) -> c_11():15 -->_1 a__take#(X1,X2) -> c_10():14 -->_3 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):5 -->_2 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):5 -->_3 mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)):4 -->_2 mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)):4 -->_3 mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)):3 -->_2 mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)):3 -->_3 mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)):2 -->_2 mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)):2 6:W:a__eq#(X,Y) -> c_1() 7:W:a__eq#(X1,X2) -> c_2() 8:W:a__eq#(0(),0()) -> c_3() 9:W:a__inf#(X) -> c_5() 10:W:a__inf#(X) -> c_6() 11:W:a__length#(X) -> c_7() 12:W:a__length#(cons(X,L)) -> c_8() 13:W:a__length#(nil()) -> c_9() 14:W:a__take#(X1,X2) -> c_10() 15:W:a__take#(0(),X) -> c_11() 16:W:a__take#(s(X),cons(Y,L)) -> c_12() 17:W:mark#(0()) -> c_13() 18:W:mark#(cons(X1,X2)) -> c_14() 19:W:mark#(false()) -> c_16() 20:W:mark#(nil()) -> c_19() 21:W:mark#(s(X)) -> c_20() 22:W:mark#(true()) -> c_22() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: a__inf#(X) -> c_5() 10: a__inf#(X) -> c_6() 11: a__length#(X) -> c_7() 12: a__length#(cons(X,L)) -> c_8() 13: a__length#(nil()) -> c_9() 14: a__take#(X1,X2) -> c_10() 15: a__take#(0(),X) -> c_11() 16: a__take#(s(X),cons(Y,L)) -> c_12() 17: mark#(0()) -> c_13() 18: mark#(cons(X1,X2)) -> c_14() 19: mark#(false()) -> c_16() 20: mark#(nil()) -> c_19() 21: mark#(s(X)) -> c_20() 22: mark#(true()) -> c_22() 6: a__eq#(X,Y) -> c_1() 7: a__eq#(X1,X2) -> c_2() 8: a__eq#(0(),0()) -> c_3() * Step 4: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)) mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)) mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak TRS: a__eq(X,Y) -> false() a__eq(X1,X2) -> eq(X1,X2) a__eq(0(),0()) -> true() a__eq(s(X),s(Y)) -> a__eq(X,Y) a__inf(X) -> cons(X,inf(s(X))) a__inf(X) -> inf(X) a__length(X) -> length(X) a__length(cons(X,L)) -> s(length(L)) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),X) -> nil() a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(eq(X1,X2)) -> a__eq(X1,X2) mark(false()) -> false() mark(inf(X)) -> a__inf(mark(X)) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(true()) -> true() - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1,a__eq#/2,a__inf#/1,a__length#/1,a__take#/2,mark#/1} / {0/0 ,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0,c_1/0,c_2/0,c_3/0,c_4/1,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_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/2,c_19/0,c_20/0,c_21/3,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq#,a__inf#,a__length#,a__take# ,mark#} and constructors {0,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) -->_1 a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)):1 2:S:mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) -->_1 a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)):1 3:S:mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)) -->_2 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):5 -->_2 mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)):4 -->_2 mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)):3 -->_2 mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)):2 4:S:mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)) -->_2 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):5 -->_2 mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)):4 -->_2 mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)):3 -->_2 mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)):2 5:S:mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):5 -->_2 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):5 -->_3 mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)):4 -->_2 mark#(length(X)) -> c_18(a__length#(mark(X)),mark#(X)):4 -->_3 mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)):3 -->_2 mark#(inf(X)) -> c_17(a__inf#(mark(X)),mark#(X)):3 -->_3 mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)):2 -->_2 mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mark#(inf(X)) -> c_17(mark#(X)) mark#(length(X)) -> c_18(mark#(X)) mark#(take(X1,X2)) -> c_21(mark#(X1),mark#(X2)) * Step 5: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) mark#(inf(X)) -> c_17(mark#(X)) mark#(length(X)) -> c_18(mark#(X)) mark#(take(X1,X2)) -> c_21(mark#(X1),mark#(X2)) - Weak TRS: a__eq(X,Y) -> false() a__eq(X1,X2) -> eq(X1,X2) a__eq(0(),0()) -> true() a__eq(s(X),s(Y)) -> a__eq(X,Y) a__inf(X) -> cons(X,inf(s(X))) a__inf(X) -> inf(X) a__length(X) -> length(X) a__length(cons(X,L)) -> s(length(L)) a__length(nil()) -> 0() a__take(X1,X2) -> take(X1,X2) a__take(0(),X) -> nil() a__take(s(X),cons(Y,L)) -> cons(Y,take(X,L)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(X1,X2) mark(eq(X1,X2)) -> a__eq(X1,X2) mark(false()) -> false() mark(inf(X)) -> a__inf(mark(X)) mark(length(X)) -> a__length(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(true()) -> true() - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1,a__eq#/2,a__inf#/1,a__length#/1,a__take#/2,mark#/1} / {0/0 ,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0,c_1/0,c_2/0,c_3/0,c_4/1,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_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/1,c_19/0,c_20/0,c_21/2,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq#,a__inf#,a__length#,a__take# ,mark#} and constructors {0,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) mark#(inf(X)) -> c_17(mark#(X)) mark#(length(X)) -> c_18(mark#(X)) mark#(take(X1,X2)) -> c_21(mark#(X1),mark#(X2)) * Step 6: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) mark#(inf(X)) -> c_17(mark#(X)) mark#(length(X)) -> c_18(mark#(X)) mark#(take(X1,X2)) -> c_21(mark#(X1),mark#(X2)) - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1,a__eq#/2,a__inf#/1,a__length#/1,a__take#/2,mark#/1} / {0/0 ,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0,c_1/0,c_2/0,c_3/0,c_4/1,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_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/1,c_19/0,c_20/0,c_21/2,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq#,a__inf#,a__length#,a__take# ,mark#} and constructors {0,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component mark#(inf(X)) -> c_17(mark#(X)) mark#(length(X)) -> c_18(mark#(X)) mark#(take(X1,X2)) -> c_21(mark#(X1),mark#(X2)) and a lower component a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) Further, following extension rules are added to the lower component. mark#(inf(X)) -> mark#(X) mark#(length(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) ** Step 6.a:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(inf(X)) -> c_17(mark#(X)) mark#(length(X)) -> c_18(mark#(X)) mark#(take(X1,X2)) -> c_21(mark#(X1),mark#(X2)) - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1,a__eq#/2,a__inf#/1,a__length#/1,a__take#/2,mark#/1} / {0/0 ,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0,c_1/0,c_2/0,c_3/0,c_4/1,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_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/1,c_19/0,c_20/0,c_21/2,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq#,a__inf#,a__length#,a__take# ,mark#} and constructors {0,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, 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: The following argument positions are considered usable: uargs(c_17) = {1}, uargs(c_18) = {1}, uargs(c_21) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__eq) = [0] p(a__inf) = [0] p(a__length) = [0] p(a__take) = [0] p(cons) = [1] x1 + [1] x2 + [0] p(eq) = [1] x1 + [1] x2 + [0] p(false) = [0] p(inf) = [1] x1 + [0] p(length) = [1] x1 + [0] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [0] p(take) = [1] x1 + [1] x2 + [6] p(true) = [0] p(a__eq#) = [0] p(a__inf#) = [0] p(a__length#) = [0] p(a__take#) = [0] p(mark#) = [4] x1 + [3] 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) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [1] x1 + [0] p(c_18) = [1] x1 + [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [1] x1 + [1] x2 + [10] p(c_22) = [0] Following rules are strictly oriented: mark#(take(X1,X2)) = [4] X1 + [4] X2 + [27] > [4] X1 + [4] X2 + [16] = c_21(mark#(X1),mark#(X2)) Following rules are (at-least) weakly oriented: mark#(inf(X)) = [4] X + [3] >= [4] X + [3] = c_17(mark#(X)) mark#(length(X)) = [4] X + [3] >= [4] X + [3] = c_18(mark#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(inf(X)) -> c_17(mark#(X)) mark#(length(X)) -> c_18(mark#(X)) - Weak DPs: mark#(take(X1,X2)) -> c_21(mark#(X1),mark#(X2)) - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1,a__eq#/2,a__inf#/1,a__length#/1,a__take#/2,mark#/1} / {0/0 ,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0,c_1/0,c_2/0,c_3/0,c_4/1,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_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/1,c_19/0,c_20/0,c_21/2,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq#,a__inf#,a__length#,a__take# ,mark#} and constructors {0,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, 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: The following argument positions are considered usable: uargs(c_17) = {1}, uargs(c_18) = {1}, uargs(c_21) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__eq) = [0] p(a__inf) = [0] p(a__length) = [0] p(a__take) = [0] p(cons) = [1] x1 + [1] x2 + [0] p(eq) = [1] x1 + [1] x2 + [0] p(false) = [0] p(inf) = [1] x1 + [0] p(length) = [1] x1 + [1] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [0] p(take) = [1] x1 + [1] x2 + [0] p(true) = [0] p(a__eq#) = [0] p(a__inf#) = [0] p(a__length#) = [0] p(a__take#) = [0] p(mark#) = [5] x1 + [0] 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) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [1] x1 + [0] p(c_18) = [1] x1 + [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [1] x1 + [1] x2 + [0] p(c_22) = [0] Following rules are strictly oriented: mark#(length(X)) = [5] X + [5] > [5] X + [0] = c_18(mark#(X)) Following rules are (at-least) weakly oriented: mark#(inf(X)) = [5] X + [0] >= [5] X + [0] = c_17(mark#(X)) mark#(take(X1,X2)) = [5] X1 + [5] X2 + [0] >= [5] X1 + [5] X2 + [0] = c_21(mark#(X1),mark#(X2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(inf(X)) -> c_17(mark#(X)) - Weak DPs: mark#(length(X)) -> c_18(mark#(X)) mark#(take(X1,X2)) -> c_21(mark#(X1),mark#(X2)) - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1,a__eq#/2,a__inf#/1,a__length#/1,a__take#/2,mark#/1} / {0/0 ,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0,c_1/0,c_2/0,c_3/0,c_4/1,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_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/1,c_19/0,c_20/0,c_21/2,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq#,a__inf#,a__length#,a__take# ,mark#} and constructors {0,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, 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: The following argument positions are considered usable: uargs(c_17) = {1}, uargs(c_18) = {1}, uargs(c_21) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__eq) = [0] p(a__inf) = [0] p(a__length) = [0] p(a__take) = [0] p(cons) = [1] x1 + [1] x2 + [0] p(eq) = [1] x1 + [1] x2 + [0] p(false) = [0] p(inf) = [1] x1 + [1] p(length) = [1] x1 + [0] p(mark) = [0] p(nil) = [0] p(s) = [1] x1 + [0] p(take) = [1] x1 + [1] x2 + [2] p(true) = [0] p(a__eq#) = [0] p(a__inf#) = [0] p(a__length#) = [0] p(a__take#) = [0] p(mark#) = [5] x1 + [10] 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) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [1] x1 + [0] p(c_18) = [1] x1 + [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [1] x1 + [1] x2 + [0] p(c_22) = [0] Following rules are strictly oriented: mark#(inf(X)) = [5] X + [15] > [5] X + [10] = c_17(mark#(X)) Following rules are (at-least) weakly oriented: mark#(length(X)) = [5] X + [10] >= [5] X + [10] = c_18(mark#(X)) mark#(take(X1,X2)) = [5] X1 + [5] X2 + [20] >= [5] X1 + [5] X2 + [20] = c_21(mark#(X1),mark#(X2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mark#(inf(X)) -> c_17(mark#(X)) mark#(length(X)) -> c_18(mark#(X)) mark#(take(X1,X2)) -> c_21(mark#(X1),mark#(X2)) - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1,a__eq#/2,a__inf#/1,a__length#/1,a__take#/2,mark#/1} / {0/0 ,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0,c_1/0,c_2/0,c_3/0,c_4/1,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_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/1,c_19/0,c_20/0,c_21/2,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq#,a__inf#,a__length#,a__take# ,mark#} and constructors {0,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) - Weak DPs: mark#(inf(X)) -> mark#(X) mark#(length(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1,a__eq#/2,a__inf#/1,a__length#/1,a__take#/2,mark#/1} / {0/0 ,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0,c_1/0,c_2/0,c_3/0,c_4/1,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_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/1,c_19/0,c_20/0,c_21/2,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq#,a__inf#,a__length#,a__take# ,mark#} and constructors {0,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_4) = {1}, uargs(c_15) = {1} Following symbols are considered usable: {a__eq#,a__inf#,a__length#,a__take#,mark#} TcT has computed the following interpretation: p(0) = [0] p(a__eq) = [0] p(a__inf) = [0] p(a__length) = [0] p(a__take) = [0] p(cons) = [0] p(eq) = [0] p(false) = [0] p(inf) = [0] p(length) = [0] p(mark) = [1] x1 + [0] p(nil) = [0] p(s) = [0] p(take) = [0] p(true) = [0] p(a__eq#) = [1] p(a__inf#) = [1] x1 + [0] p(a__length#) = [0] p(a__take#) = [1] x2 + [0] p(mark#) = [10] p(c_1) = [0] p(c_2) = [0] p(c_3) = [2] p(c_4) = [1] x1 + [0] p(c_5) = [2] p(c_6) = [0] p(c_7) = [1] p(c_8) = [2] p(c_9) = [0] p(c_10) = [2] p(c_11) = [2] p(c_12) = [2] p(c_13) = [0] p(c_14) = [1] p(c_15) = [1] x1 + [7] p(c_16) = [1] p(c_17) = [1] x1 + [1] p(c_18) = [0] p(c_19) = [1] p(c_20) = [4] p(c_21) = [1] x2 + [0] p(c_22) = [1] Following rules are strictly oriented: mark#(eq(X1,X2)) = [10] > [8] = c_15(a__eq#(X1,X2)) Following rules are (at-least) weakly oriented: a__eq#(s(X),s(Y)) = [1] >= [1] = c_4(a__eq#(X,Y)) mark#(inf(X)) = [10] >= [10] = mark#(X) mark#(length(X)) = [10] >= [10] = mark#(X) mark#(take(X1,X2)) = [10] >= [10] = mark#(X1) mark#(take(X1,X2)) = [10] >= [10] = mark#(X2) ** Step 6.b:2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) - Weak DPs: mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) mark#(inf(X)) -> mark#(X) mark#(length(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1,a__eq#/2,a__inf#/1,a__length#/1,a__take#/2,mark#/1} / {0/0 ,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0,c_1/0,c_2/0,c_3/0,c_4/1,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_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/1,c_19/0,c_20/0,c_21/2,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq#,a__inf#,a__length#,a__take# ,mark#} and constructors {0,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_4) = {1}, uargs(c_15) = {1} Following symbols are considered usable: {a__eq#,a__inf#,a__length#,a__take#,mark#} TcT has computed the following interpretation: p(0) = [0] p(a__eq) = [4] x2 + [0] p(a__inf) = [1] x1 + [1] p(a__length) = [4] x1 + [0] p(a__take) = [2] x1 + [1] x2 + [8] p(cons) = [8] p(eq) = [1] x1 + [0] p(false) = [0] p(inf) = [1] x1 + [0] p(length) = [1] x1 + [2] p(mark) = [1] x1 + [1] p(nil) = [0] p(s) = [1] x1 + [8] p(take) = [1] x1 + [1] x2 + [8] p(true) = [0] p(a__eq#) = [1] x1 + [0] p(a__inf#) = [0] p(a__length#) = [1] x1 + [2] p(a__take#) = [1] x1 + [0] p(mark#) = [1] x1 + [8] p(c_1) = [2] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] x1 + [0] p(c_5) = [1] p(c_6) = [1] p(c_7) = [1] p(c_8) = [0] p(c_9) = [0] p(c_10) = [2] p(c_11) = [0] p(c_12) = [1] p(c_13) = [2] p(c_14) = [1] p(c_15) = [1] x1 + [8] p(c_16) = [0] p(c_17) = [1] x1 + [1] p(c_18) = [1] x1 + [1] p(c_19) = [2] p(c_20) = [1] p(c_21) = [1] x1 + [0] p(c_22) = [1] Following rules are strictly oriented: a__eq#(s(X),s(Y)) = [1] X + [8] > [1] X + [0] = c_4(a__eq#(X,Y)) Following rules are (at-least) weakly oriented: mark#(eq(X1,X2)) = [1] X1 + [8] >= [1] X1 + [8] = c_15(a__eq#(X1,X2)) mark#(inf(X)) = [1] X + [8] >= [1] X + [8] = mark#(X) mark#(length(X)) = [1] X + [10] >= [1] X + [8] = mark#(X) mark#(take(X1,X2)) = [1] X1 + [1] X2 + [16] >= [1] X1 + [8] = mark#(X1) mark#(take(X1,X2)) = [1] X1 + [1] X2 + [16] >= [1] X2 + [8] = mark#(X2) ** Step 6.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) mark#(inf(X)) -> mark#(X) mark#(length(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) - Signature: {a__eq/2,a__inf/1,a__length/1,a__take/2,mark/1,a__eq#/2,a__inf#/1,a__length#/1,a__take#/2,mark#/1} / {0/0 ,cons/2,eq/2,false/0,inf/1,length/1,nil/0,s/1,take/2,true/0,c_1/0,c_2/0,c_3/0,c_4/1,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_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/1,c_19/0,c_20/0,c_21/2,c_22/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__eq#,a__inf#,a__length#,a__take# ,mark#} and constructors {0,cons,eq,false,inf,length,nil,s,take,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))