MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { minus(X1, X2) -> n__minus(X1, X2) , minus(n__0(), Y) -> 0() , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) , 0() -> n__0() , activate(X) -> X , activate(n__0()) -> 0() , activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) , activate(n__minus(X1, X2)) -> minus(X1, X2) , geq(X, n__0()) -> true() , geq(n__0(), n__s(Y)) -> false() , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) , div(X1, X2) -> n__div(X1, X2) , div(0(), n__s(Y)) -> 0() , div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0()) , s(X) -> n__s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { minus^#(X1, X2) -> c_1(X1, X2) , minus^#(n__0(), Y) -> c_2(0^#()) , minus^#(n__s(X), n__s(Y)) -> c_3(minus^#(activate(X), activate(Y))) , 0^#() -> c_4() , activate^#(X) -> c_5(X) , activate^#(n__0()) -> c_6(0^#()) , activate^#(n__s(X)) -> c_7(s^#(activate(X))) , activate^#(n__div(X1, X2)) -> c_8(div^#(activate(X1), X2)) , activate^#(n__minus(X1, X2)) -> c_9(minus^#(X1, X2)) , s^#(X) -> c_16(X) , div^#(X1, X2) -> c_13(X1, X2) , div^#(0(), n__s(Y)) -> c_14(0^#()) , div^#(s(X), n__s(Y)) -> c_15(if^#(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0())) , geq^#(X, n__0()) -> c_10() , geq^#(n__0(), n__s(Y)) -> c_11() , geq^#(n__s(X), n__s(Y)) -> c_12(geq^#(activate(X), activate(Y))) , if^#(true(), X, Y) -> c_17(activate^#(X)) , if^#(false(), X, Y) -> c_18(activate^#(Y)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(X1, X2) -> c_1(X1, X2) , minus^#(n__0(), Y) -> c_2(0^#()) , minus^#(n__s(X), n__s(Y)) -> c_3(minus^#(activate(X), activate(Y))) , 0^#() -> c_4() , activate^#(X) -> c_5(X) , activate^#(n__0()) -> c_6(0^#()) , activate^#(n__s(X)) -> c_7(s^#(activate(X))) , activate^#(n__div(X1, X2)) -> c_8(div^#(activate(X1), X2)) , activate^#(n__minus(X1, X2)) -> c_9(minus^#(X1, X2)) , s^#(X) -> c_16(X) , div^#(X1, X2) -> c_13(X1, X2) , div^#(0(), n__s(Y)) -> c_14(0^#()) , div^#(s(X), n__s(Y)) -> c_15(if^#(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0())) , geq^#(X, n__0()) -> c_10() , geq^#(n__0(), n__s(Y)) -> c_11() , geq^#(n__s(X), n__s(Y)) -> c_12(geq^#(activate(X), activate(Y))) , if^#(true(), X, Y) -> c_17(activate^#(X)) , if^#(false(), X, Y) -> c_18(activate^#(Y)) } Strict Trs: { minus(X1, X2) -> n__minus(X1, X2) , minus(n__0(), Y) -> 0() , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) , 0() -> n__0() , activate(X) -> X , activate(n__0()) -> 0() , activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) , activate(n__minus(X1, X2)) -> minus(X1, X2) , geq(X, n__0()) -> true() , geq(n__0(), n__s(Y)) -> false() , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) , div(X1, X2) -> n__div(X1, X2) , div(0(), n__s(Y)) -> 0() , div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0()) , s(X) -> n__s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {4,14,15} by applications of Pre({4,14,15}) = {1,2,5,6,10,11,12,16}. Here rules are labeled as follows: DPs: { 1: minus^#(X1, X2) -> c_1(X1, X2) , 2: minus^#(n__0(), Y) -> c_2(0^#()) , 3: minus^#(n__s(X), n__s(Y)) -> c_3(minus^#(activate(X), activate(Y))) , 4: 0^#() -> c_4() , 5: activate^#(X) -> c_5(X) , 6: activate^#(n__0()) -> c_6(0^#()) , 7: activate^#(n__s(X)) -> c_7(s^#(activate(X))) , 8: activate^#(n__div(X1, X2)) -> c_8(div^#(activate(X1), X2)) , 9: activate^#(n__minus(X1, X2)) -> c_9(minus^#(X1, X2)) , 10: s^#(X) -> c_16(X) , 11: div^#(X1, X2) -> c_13(X1, X2) , 12: div^#(0(), n__s(Y)) -> c_14(0^#()) , 13: div^#(s(X), n__s(Y)) -> c_15(if^#(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0())) , 14: geq^#(X, n__0()) -> c_10() , 15: geq^#(n__0(), n__s(Y)) -> c_11() , 16: geq^#(n__s(X), n__s(Y)) -> c_12(geq^#(activate(X), activate(Y))) , 17: if^#(true(), X, Y) -> c_17(activate^#(X)) , 18: if^#(false(), X, Y) -> c_18(activate^#(Y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(X1, X2) -> c_1(X1, X2) , minus^#(n__0(), Y) -> c_2(0^#()) , minus^#(n__s(X), n__s(Y)) -> c_3(minus^#(activate(X), activate(Y))) , activate^#(X) -> c_5(X) , activate^#(n__0()) -> c_6(0^#()) , activate^#(n__s(X)) -> c_7(s^#(activate(X))) , activate^#(n__div(X1, X2)) -> c_8(div^#(activate(X1), X2)) , activate^#(n__minus(X1, X2)) -> c_9(minus^#(X1, X2)) , s^#(X) -> c_16(X) , div^#(X1, X2) -> c_13(X1, X2) , div^#(0(), n__s(Y)) -> c_14(0^#()) , div^#(s(X), n__s(Y)) -> c_15(if^#(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0())) , geq^#(n__s(X), n__s(Y)) -> c_12(geq^#(activate(X), activate(Y))) , if^#(true(), X, Y) -> c_17(activate^#(X)) , if^#(false(), X, Y) -> c_18(activate^#(Y)) } Strict Trs: { minus(X1, X2) -> n__minus(X1, X2) , minus(n__0(), Y) -> 0() , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) , 0() -> n__0() , activate(X) -> X , activate(n__0()) -> 0() , activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) , activate(n__minus(X1, X2)) -> minus(X1, X2) , geq(X, n__0()) -> true() , geq(n__0(), n__s(Y)) -> false() , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) , div(X1, X2) -> n__div(X1, X2) , div(0(), n__s(Y)) -> 0() , div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0()) , s(X) -> n__s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) } Weak DPs: { 0^#() -> c_4() , geq^#(X, n__0()) -> c_10() , geq^#(n__0(), n__s(Y)) -> c_11() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,5,11} by applications of Pre({2,5,11}) = {1,3,4,7,8,9,10,14,15}. Here rules are labeled as follows: DPs: { 1: minus^#(X1, X2) -> c_1(X1, X2) , 2: minus^#(n__0(), Y) -> c_2(0^#()) , 3: minus^#(n__s(X), n__s(Y)) -> c_3(minus^#(activate(X), activate(Y))) , 4: activate^#(X) -> c_5(X) , 5: activate^#(n__0()) -> c_6(0^#()) , 6: activate^#(n__s(X)) -> c_7(s^#(activate(X))) , 7: activate^#(n__div(X1, X2)) -> c_8(div^#(activate(X1), X2)) , 8: activate^#(n__minus(X1, X2)) -> c_9(minus^#(X1, X2)) , 9: s^#(X) -> c_16(X) , 10: div^#(X1, X2) -> c_13(X1, X2) , 11: div^#(0(), n__s(Y)) -> c_14(0^#()) , 12: div^#(s(X), n__s(Y)) -> c_15(if^#(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0())) , 13: geq^#(n__s(X), n__s(Y)) -> c_12(geq^#(activate(X), activate(Y))) , 14: if^#(true(), X, Y) -> c_17(activate^#(X)) , 15: if^#(false(), X, Y) -> c_18(activate^#(Y)) , 16: 0^#() -> c_4() , 17: geq^#(X, n__0()) -> c_10() , 18: geq^#(n__0(), n__s(Y)) -> c_11() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(X1, X2) -> c_1(X1, X2) , minus^#(n__s(X), n__s(Y)) -> c_3(minus^#(activate(X), activate(Y))) , activate^#(X) -> c_5(X) , activate^#(n__s(X)) -> c_7(s^#(activate(X))) , activate^#(n__div(X1, X2)) -> c_8(div^#(activate(X1), X2)) , activate^#(n__minus(X1, X2)) -> c_9(minus^#(X1, X2)) , s^#(X) -> c_16(X) , div^#(X1, X2) -> c_13(X1, X2) , div^#(s(X), n__s(Y)) -> c_15(if^#(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0())) , geq^#(n__s(X), n__s(Y)) -> c_12(geq^#(activate(X), activate(Y))) , if^#(true(), X, Y) -> c_17(activate^#(X)) , if^#(false(), X, Y) -> c_18(activate^#(Y)) } Strict Trs: { minus(X1, X2) -> n__minus(X1, X2) , minus(n__0(), Y) -> 0() , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) , 0() -> n__0() , activate(X) -> X , activate(n__0()) -> 0() , activate(n__s(X)) -> s(activate(X)) , activate(n__div(X1, X2)) -> div(activate(X1), X2) , activate(n__minus(X1, X2)) -> minus(X1, X2) , geq(X, n__0()) -> true() , geq(n__0(), n__s(Y)) -> false() , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) , div(X1, X2) -> n__div(X1, X2) , div(0(), n__s(Y)) -> 0() , div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0()) , s(X) -> n__s(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) } Weak DPs: { minus^#(n__0(), Y) -> c_2(0^#()) , 0^#() -> c_4() , activate^#(n__0()) -> c_6(0^#()) , div^#(0(), n__s(Y)) -> c_14(0^#()) , geq^#(X, n__0()) -> c_10() , geq^#(n__0(), n__s(Y)) -> c_11() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..