MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , dbls(nil()) -> nil() , dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) , sel(0(), cons(X, Y)) -> X , sel(s(X), cons(Y, Z)) -> sel(X, Z) , indx(nil(), X) -> nil() , indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) , from(X) -> cons(X, from(s(X))) } Obligation: innermost runtime complexity Answer: MAYBE We add following weak dependency pairs: Strict DPs: { dbl^#(0()) -> c_1() , dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(nil()) -> c_3() , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , sel^#(0(), cons(X, Y)) -> c_5() , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , indx^#(nil(), X) -> c_7() , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(X))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { dbl^#(0()) -> c_1() , dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(nil()) -> c_3() , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , sel^#(0(), cons(X, Y)) -> c_5() , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , indx^#(nil(), X) -> c_7() , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(X))) } Strict Trs: { dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , dbls(nil()) -> nil() , dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) , sel(0(), cons(X, Y)) -> X , sel(s(X), cons(Y, Z)) -> sel(X, Z) , indx(nil(), X) -> nil() , indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) , from(X) -> cons(X, from(s(X))) } Obligation: innermost runtime complexity Answer: MAYBE No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { dbl^#(0()) -> c_1() , dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(nil()) -> c_3() , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , sel^#(0(), cons(X, Y)) -> c_5() , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , indx^#(nil(), X) -> c_7() , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(X))) } Obligation: innermost runtime complexity Answer: MAYBE The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(c_2) = {1}, Uargs(c_4) = {1, 2}, Uargs(c_6) = {1}, Uargs(c_8) = {1, 2}, Uargs(c_9) = {1} TcT has computed following constructor-restricted matrix interpretation. [0] = [0] [s](x1) = [1] x1 + [0] [nil] = [0] [cons](x1, x2) = [1] x1 + [1] x2 + [0] [dbl^#](x1) = [0] [c_1] = [1] [c_2](x1) = [1] x1 + [1] [dbls^#](x1) = [0] [c_3] = [1] [c_4](x1, x2) = [1] x1 + [1] x2 + [1] [sel^#](x1, x2) = [0] [c_5] = [1] [c_6](x1) = [1] x1 + [1] [indx^#](x1, x2) = [2] x2 + [1] [c_7] = [0] [c_8](x1, x2) = [1] x1 + [1] x2 + [0] [from^#](x1) = [2] x1 + [2] [c_9](x1) = [1] x1 + [2] This order satisfies following ordering constraints: Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { dbl^#(0()) -> c_1() , dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(nil()) -> c_3() , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , sel^#(0(), cons(X, Y)) -> c_5() , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(X))) } Weak DPs: { indx^#(nil(), X) -> c_7() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,3,5} by applications of Pre({1,3,5}) = {2,4,6,7}. Here rules are labeled as follows: DPs: { 1: dbl^#(0()) -> c_1() , 2: dbl^#(s(X)) -> c_2(dbl^#(X)) , 3: dbls^#(nil()) -> c_3() , 4: dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , 5: sel^#(0(), cons(X, Y)) -> c_5() , 6: sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , 7: indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , 8: from^#(X) -> c_9(from^#(s(X))) , 9: indx^#(nil(), X) -> c_7() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(X))) } Weak DPs: { dbl^#(0()) -> c_1() , dbls^#(nil()) -> c_3() , sel^#(0(), cons(X, Y)) -> c_5() , indx^#(nil(), X) -> c_7() } Obligation: innermost runtime complexity Answer: MAYBE The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { dbl^#(0()) -> c_1() , dbls^#(nil()) -> c_3() , sel^#(0(), cons(X, Y)) -> c_5() , indx^#(nil(), X) -> c_7() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { dbl^#(s(X)) -> c_2(dbl^#(X)) , dbls^#(cons(X, Y)) -> c_4(dbl^#(X), dbls^#(Y)) , sel^#(s(X), cons(Y, Z)) -> c_6(sel^#(X, Z)) , indx^#(cons(X, Y), Z) -> c_8(sel^#(X, Z), indx^#(Y, Z)) , from^#(X) -> c_9(from^#(s(X))) } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrices' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrix interpretation of dimension 4' failed due to the following reason: The input cannot be shown compatible 2) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 3) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 4) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 5) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 6) 'matrix interpretation of dimension 1' failed due to the following reason: The input cannot be shown compatible 2) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. Arrrr..