WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + 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_ = WIDP} + Details: We add the following weak innermost dependency pairs: 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#(length(X)) -> c_18(a__length#(mark(X))) mark#(nil()) -> c_19() mark#(s(X)) -> c_20() mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2))) mark#(true()) -> c_22() Weak DPs and mark the set of starting terms. * Step 2: WeightGap WORST_CASE(?,O(n^1)) + 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#(length(X)) -> c_18(a__length#(mark(X))) mark#(nil()) -> c_19() mark#(s(X)) -> c_20() mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2))) mark#(true()) -> c_22() - 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,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_21/1,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 = WgOnTrs} + 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(a__inf) = {1}, uargs(a__length) = {1}, uargs(a__take) = {1,2}, uargs(a__inf#) = {1}, uargs(a__length#) = {1}, uargs(a__take#) = {1,2}, uargs(c_4) = {1}, uargs(c_15) = {1}, uargs(c_17) = {1}, uargs(c_18) = {1}, uargs(c_21) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(a__eq) = [1] x1 + [2] p(a__inf) = [1] x1 + [5] p(a__length) = [1] x1 + [5] p(a__take) = [1] x1 + [1] x2 + [5] p(cons) = [1] x2 + [1] p(eq) = [1] x1 + [1] p(false) = [1] p(inf) = [1] x1 + [2] p(length) = [1] x1 + [3] p(mark) = [3] x1 + [0] p(nil) = [1] p(s) = [1] x1 + [1] p(take) = [1] x1 + [1] x2 + [4] p(true) = [2] p(a__eq#) = [4] p(a__inf#) = [1] x1 + [4] p(a__length#) = [1] x1 + [7] p(a__take#) = [1] x1 + [1] x2 + [1] p(mark#) = [3] x1 + [2] p(c_1) = [0] p(c_2) = [1] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [4] p(c_6) = [2] p(c_7) = [2] p(c_8) = [1] p(c_9) = [4] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] p(c_13) = [1] p(c_14) = [0] p(c_15) = [1] x1 + [0] p(c_16) = [1] p(c_17) = [1] x1 + [1] p(c_18) = [1] x1 + [0] p(c_19) = [4] p(c_20) = [0] p(c_21) = [1] x1 + [0] p(c_22) = [0] Following rules are strictly oriented: a__eq#(X,Y) = [4] > [0] = c_1() a__eq#(X1,X2) = [4] > [1] = c_2() a__eq#(0(),0()) = [4] > [0] = c_3() a__inf#(X) = [1] X + [4] > [2] = c_6() a__length#(X) = [1] X + [7] > [2] = c_7() a__length#(cons(X,L)) = [1] L + [8] > [1] = c_8() a__length#(nil()) = [8] > [4] = c_9() a__take#(X1,X2) = [1] X1 + [1] X2 + [1] > [0] = c_10() a__take#(0(),X) = [1] X + [3] > [0] = c_11() a__take#(s(X),cons(Y,L)) = [1] L + [1] X + [3] > [1] = c_12() mark#(0()) = [8] > [1] = c_13() mark#(cons(X1,X2)) = [3] X2 + [5] > [0] = c_14() mark#(eq(X1,X2)) = [3] X1 + [5] > [4] = c_15(a__eq#(X1,X2)) mark#(false()) = [5] > [1] = c_16() mark#(inf(X)) = [3] X + [8] > [3] X + [5] = c_17(a__inf#(mark(X))) mark#(length(X)) = [3] X + [11] > [3] X + [7] = c_18(a__length#(mark(X))) mark#(nil()) = [5] > [4] = c_19() mark#(s(X)) = [3] X + [5] > [0] = c_20() mark#(take(X1,X2)) = [3] X1 + [3] X2 + [14] > [3] X1 + [3] X2 + [1] = c_21(a__take#(mark(X1),mark(X2))) mark#(true()) = [8] > [0] = c_22() a__eq(X,Y) = [1] X + [2] > [1] = false() a__eq(X1,X2) = [1] X1 + [2] > [1] X1 + [1] = eq(X1,X2) a__eq(0(),0()) = [4] > [2] = true() a__eq(s(X),s(Y)) = [1] X + [3] > [1] X + [2] = a__eq(X,Y) a__inf(X) = [1] X + [5] > [1] X + [4] = cons(X,inf(s(X))) a__inf(X) = [1] X + [5] > [1] X + [2] = inf(X) a__length(X) = [1] X + [5] > [1] X + [3] = length(X) a__length(cons(X,L)) = [1] L + [6] > [1] L + [4] = s(length(L)) a__length(nil()) = [6] > [2] = 0() a__take(X1,X2) = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [4] = take(X1,X2) a__take(0(),X) = [1] X + [7] > [1] = nil() a__take(s(X),cons(Y,L)) = [1] L + [1] X + [7] > [1] L + [1] X + [5] = cons(Y,take(X,L)) mark(0()) = [6] > [2] = 0() mark(cons(X1,X2)) = [3] X2 + [3] > [1] X2 + [1] = cons(X1,X2) mark(eq(X1,X2)) = [3] X1 + [3] > [1] X1 + [2] = a__eq(X1,X2) mark(false()) = [3] > [1] = false() mark(inf(X)) = [3] X + [6] > [3] X + [5] = a__inf(mark(X)) mark(length(X)) = [3] X + [9] > [3] X + [5] = a__length(mark(X)) mark(nil()) = [3] > [1] = nil() mark(s(X)) = [3] X + [3] > [1] X + [1] = s(X) mark(take(X1,X2)) = [3] X1 + [3] X2 + [12] > [3] X1 + [3] X2 + [5] = a__take(mark(X1),mark(X2)) mark(true()) = [6] > [2] = true() Following rules are (at-least) weakly oriented: a__eq#(s(X),s(Y)) = [4] >= [4] = c_4(a__eq#(X,Y)) a__inf#(X) = [1] X + [4] >= [4] = c_5() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) a__inf#(X) -> c_5() - Weak DPs: a__eq#(X,Y) -> c_1() a__eq#(X1,X2) -> c_2() a__eq#(0(),0()) -> c_3() 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#(length(X)) -> c_18(a__length#(mark(X))) mark#(nil()) -> c_19() mark#(s(X)) -> c_20() mark#(take(X1,X2)) -> c_21(a__take#(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_17/1,c_18/1,c_19/0,c_20/0,c_21/1,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():5 -->_1 a__eq#(X1,X2) -> c_2():4 -->_1 a__eq#(X,Y) -> c_1():3 -->_1 a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)):1 2:S:a__inf#(X) -> c_5() 3:W:a__eq#(X,Y) -> c_1() 4:W:a__eq#(X1,X2) -> c_2() 5:W:a__eq#(0(),0()) -> c_3() 6:W:a__inf#(X) -> c_6() 7:W:a__length#(X) -> c_7() 8:W:a__length#(cons(X,L)) -> c_8() 9:W:a__length#(nil()) -> c_9() 10:W:a__take#(X1,X2) -> c_10() 11:W:a__take#(0(),X) -> c_11() 12:W:a__take#(s(X),cons(Y,L)) -> c_12() 13:W:mark#(0()) -> c_13() 14:W:mark#(cons(X1,X2)) -> c_14() 15:W:mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) -->_1 a__eq#(0(),0()) -> c_3():5 -->_1 a__eq#(X1,X2) -> c_2():4 -->_1 a__eq#(X,Y) -> c_1():3 -->_1 a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)):1 16:W:mark#(false()) -> c_16() 17:W:mark#(inf(X)) -> c_17(a__inf#(mark(X))) -->_1 a__inf#(X) -> c_6():6 -->_1 a__inf#(X) -> c_5():2 18:W:mark#(length(X)) -> c_18(a__length#(mark(X))) -->_1 a__length#(nil()) -> c_9():9 -->_1 a__length#(cons(X,L)) -> c_8():8 -->_1 a__length#(X) -> c_7():7 19:W:mark#(nil()) -> c_19() 20:W:mark#(s(X)) -> c_20() 21:W:mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2))) -->_1 a__take#(s(X),cons(Y,L)) -> c_12():12 -->_1 a__take#(0(),X) -> c_11():11 -->_1 a__take#(X1,X2) -> c_10():10 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. 22: mark#(true()) -> c_22() 21: mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2))) 20: mark#(s(X)) -> c_20() 19: mark#(nil()) -> c_19() 18: mark#(length(X)) -> c_18(a__length#(mark(X))) 16: mark#(false()) -> c_16() 14: mark#(cons(X1,X2)) -> c_14() 13: mark#(0()) -> c_13() 12: a__take#(s(X),cons(Y,L)) -> c_12() 11: a__take#(0(),X) -> c_11() 10: a__take#(X1,X2) -> c_10() 9: a__length#(nil()) -> c_9() 8: a__length#(cons(X,L)) -> c_8() 7: a__length#(X) -> c_7() 6: a__inf#(X) -> c_6() 3: a__eq#(X,Y) -> c_1() 4: a__eq#(X1,X2) -> c_2() 5: a__eq#(0(),0()) -> c_3() * Step 4: RemoveHeads WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) a__inf#(X) -> c_5() - Weak DPs: mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) mark#(inf(X)) -> c_17(a__inf#(mark(X))) - 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_21/1,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: RemoveHeads + 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:a__inf#(X) -> c_5() 15:W:mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)) -->_1 a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)):1 17:W:mark#(inf(X)) -> c_17(a__inf#(mark(X))) -->_1 a__inf#(X) -> c_5():2 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(15,mark#(eq(X1,X2)) -> c_15(a__eq#(X1,X2)))] * Step 5: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) a__inf#(X) -> c_5() - Weak DPs: mark#(inf(X)) -> c_17(a__inf#(mark(X))) - 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_21/1,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: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) - Weak DPs: a__inf#(X) -> c_5() mark#(inf(X)) -> c_17(a__inf#(mark(X))) - 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_21/1,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} Problem (S) - Strict DPs: a__inf#(X) -> c_5() - Weak DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) mark#(inf(X)) -> c_17(a__inf#(mark(X))) - 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_21/1,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} ** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) - Weak DPs: a__inf#(X) -> c_5() mark#(inf(X)) -> c_17(a__inf#(mark(X))) - 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_21/1,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#(s(X),s(Y)) -> c_4(a__eq#(X,Y)):1 2:W:a__inf#(X) -> c_5() 17:W:mark#(inf(X)) -> c_17(a__inf#(mark(X))) -->_1 a__inf#(X) -> c_5():2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 17: mark#(inf(X)) -> c_17(a__inf#(mark(X))) 2: a__inf#(X) -> c_5() ** Step 5.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) - 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_21/1,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)) ** Step 5.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) - 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_21/1,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) The strictly oriented rules are moved into the weak component. *** Step 5.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) - 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_21/1,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 first alternative for predecessorEstimation on any intersect of rules of CDG leaf and 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} Following symbols are considered usable: {a__eq#,a__inf#,a__length#,a__take#,mark#} TcT has computed the following interpretation: p(0) = [1] p(a__eq) = [1] x1 + [0] p(a__inf) = [1] x1 + [8] p(a__length) = [1] x1 + [8] p(a__take) = [2] x1 + [8] x2 + [1] p(cons) = [0] p(eq) = [1] x1 + [1] p(false) = [1] p(inf) = [1] x1 + [1] p(length) = [1] p(mark) = [1] x1 + [1] p(nil) = [2] p(s) = [1] x1 + [1] p(take) = [1] x2 + [0] p(true) = [0] p(a__eq#) = [1] x2 + [0] p(a__inf#) = [1] p(a__length#) = [2] x1 + [2] p(a__take#) = [1] x1 + [0] p(mark#) = [1] x1 + [1] p(c_1) = [0] p(c_2) = [0] 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) = [8] p(c_9) = [1] p(c_10) = [1] p(c_11) = [1] p(c_12) = [2] p(c_13) = [0] p(c_14) = [0] p(c_15) = [2] x1 + [1] p(c_16) = [1] p(c_17) = [0] p(c_18) = [2] p(c_19) = [2] p(c_20) = [1] p(c_21) = [1] x1 + [1] p(c_22) = [0] Following rules are strictly oriented: a__eq#(s(X),s(Y)) = [1] Y + [1] > [1] Y + [0] = c_4(a__eq#(X,Y)) Following rules are (at-least) weakly oriented: *** Step 5.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) - 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_21/1,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) - 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_21/1,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:W: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 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) *** Step 5.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_21/1,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 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: a__inf#(X) -> c_5() - Weak DPs: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) mark#(inf(X)) -> c_17(a__inf#(mark(X))) - 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_21/1,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__inf#(X) -> c_5() 2:W: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)):2 3:W:mark#(inf(X)) -> c_17(a__inf#(mark(X))) -->_1 a__inf#(X) -> c_5():1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: a__eq#(s(X),s(Y)) -> c_4(a__eq#(X,Y)) ** Step 5.b:2: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: a__inf#(X) -> c_5() - Weak DPs: mark#(inf(X)) -> c_17(a__inf#(mark(X))) - 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_21/1,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: Trivial + Details: Consider the dependency graph 1:S:a__inf#(X) -> c_5() 3:W:mark#(inf(X)) -> c_17(a__inf#(mark(X))) -->_1 a__inf#(X) -> c_5():1 The dependency graph contains no loops, we remove all dependency pairs. ** Step 5.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_21/1,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^1))