MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { eq() -> eq() , eq() -> true() , eq() -> false() , inf(X) -> cons() , take(0(), X) -> nil() , take(s(), cons()) -> cons() , length(cons()) -> s() , length(nil()) -> 0() } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: We add the following innermost weak dependency pairs: Strict DPs: { eq^#() -> c_1(eq^#()) , eq^#() -> c_2() , eq^#() -> c_3() , inf^#(X) -> c_4() , take^#(0(), X) -> c_5() , take^#(s(), cons()) -> c_6() , length^#(cons()) -> c_7() , length^#(nil()) -> c_8() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { eq^#() -> c_1(eq^#()) , eq^#() -> c_2() , eq^#() -> c_3() , inf^#(X) -> c_4() , take^#(0(), X) -> c_5() , take^#(s(), cons()) -> c_6() , length^#(cons()) -> c_7() , length^#(nil()) -> c_8() } Strict Trs: { eq() -> eq() , eq() -> true() , eq() -> false() , inf(X) -> cons() , take(0(), X) -> nil() , take(s(), cons()) -> cons() , length(cons()) -> s() , length(nil()) -> 0() } 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: { eq^#() -> c_1(eq^#()) , eq^#() -> c_2() , eq^#() -> c_3() , inf^#(X) -> c_4() , take^#(0(), X) -> c_5() , take^#(s(), cons()) -> c_6() , length^#(cons()) -> c_7() , length^#(nil()) -> c_8() } 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_1) = {1} TcT has computed the following constructor-restricted matrix interpretation. [cons] = [1] [1] [0] = [2] [2] [nil] = [2] [2] [s] = [1] [2] [eq^#] = [1] [1] [c_1](x1) = [1 0] x1 + [2] [0 1] [2] [c_2] = [0] [1] [c_3] = [0] [1] [inf^#](x1) = [2] [1] [c_4] = [1] [1] [take^#](x1, x2) = [1 1] x1 + [2 2] x2 + [0] [1 1] [2 2] [0] [c_5] = [1] [0] [c_6] = [2] [1] [length^#](x1) = [1 1] x1 + [2] [1 1] [2] [c_7] = [1] [0] [c_8] = [1] [1] The following symbols are considered usable {eq^#, inf^#, take^#, length^#} The order satisfies the following ordering constraints: [eq^#()] = [1] [1] ? [3] [3] = [c_1(eq^#())] [eq^#()] = [1] [1] > [0] [1] = [c_2()] [eq^#()] = [1] [1] > [0] [1] = [c_3()] [inf^#(X)] = [2] [1] > [1] [1] = [c_4()] [take^#(0(), X)] = [2 2] X + [4] [2 2] [4] > [1] [0] = [c_5()] [take^#(s(), cons())] = [7] [7] > [2] [1] = [c_6()] [length^#(cons())] = [4] [4] > [1] [0] = [c_7()] [length^#(nil())] = [6] [6] > [1] [1] = [c_8()] 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: { eq^#() -> c_1(eq^#()) } Weak DPs: { eq^#() -> c_2() , eq^#() -> c_3() , inf^#(X) -> c_4() , take^#(0(), X) -> c_5() , take^#(s(), cons()) -> c_6() , length^#(cons()) -> c_7() , length^#(nil()) -> c_8() } 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. { eq^#() -> c_2() , eq^#() -> c_3() , inf^#(X) -> c_4() , take^#(0(), X) -> c_5() , take^#(s(), cons()) -> c_6() , length^#(cons()) -> c_7() , length^#(nil()) -> c_8() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { eq^#() -> c_1(eq^#()) } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Fastest' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Fastest (timeout of 5 seconds)' failed due to the following reason: Computation stopped due to timeout after 5.0 seconds. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Polynomial Path Order (PS)' failed due to the following reason: The input cannot be shown compatible 3) 'Polynomial Path Order (PS)' failed due to the following reason: The input cannot be shown compatible 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 5.0 seconds. 2) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: We use the processor 'Small Polynomial Path Order (PS,1-bounded)' to orient following rules strictly. Trs: { eq() -> true() , eq() -> false() , inf(X) -> cons() , take(0(), X) -> nil() , take(s(), cons()) -> cons() , length(cons()) -> s() , length(nil()) -> 0() } The induced complexity on above rules (modulo remaining rules) is YES(?,O(1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The input was oriented with the instance of 'Small Polynomial Path Order (PS,1-bounded)' as induced by the safe mapping safe(eq) = {}, safe(true) = {}, safe(false) = {}, safe(inf) = {}, safe(cons) = {}, safe(take) = {}, safe(0) = {}, safe(nil) = {}, safe(s) = {}, safe(length) = {} and precedence take > inf, length > inf, take ~ length . Following symbols are considered recursive: {} The recursion depth is 0. For your convenience, here are the satisfied ordering constraints: eq() >= eq() eq() > true() eq() > false() inf(X;) > cons() take(0(), X;) > nil() take(s(), cons();) > cons() length(cons();) > s() length(nil();) > 0() We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { eq() -> eq() } Weak Trs: { eq() -> true() , eq() -> false() , inf(X) -> cons() , take(0(), X) -> nil() , take(s(), cons()) -> cons() , length(cons()) -> s() , length(nil()) -> 0() } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Fastest' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 3) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The input cannot be shown compatible 2) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The input cannot be shown compatible Arrrr..