MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { 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(X) , 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(0(), n__s(Y)) -> 0() , div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(div(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) '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 2) '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 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^#(n__0(), Y) -> c_1(0^#()) , minus^#(n__s(X), n__s(Y)) -> c_2(minus^#(activate(X), activate(Y))) , 0^#() -> c_3() , activate^#(X) -> c_4(X) , activate^#(n__0()) -> c_5(0^#()) , activate^#(n__s(X)) -> c_6(s^#(X)) , s^#(X) -> c_12(X) , geq^#(X, n__0()) -> c_7() , geq^#(n__0(), n__s(Y)) -> c_8() , geq^#(n__s(X), n__s(Y)) -> c_9(geq^#(activate(X), activate(Y))) , div^#(0(), n__s(Y)) -> c_10(0^#()) , div^#(s(X), n__s(Y)) -> c_11(if^#(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0())) , if^#(true(), X, Y) -> c_13(activate^#(X)) , if^#(false(), X, Y) -> c_14(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^#(n__0(), Y) -> c_1(0^#()) , minus^#(n__s(X), n__s(Y)) -> c_2(minus^#(activate(X), activate(Y))) , 0^#() -> c_3() , activate^#(X) -> c_4(X) , activate^#(n__0()) -> c_5(0^#()) , activate^#(n__s(X)) -> c_6(s^#(X)) , s^#(X) -> c_12(X) , geq^#(X, n__0()) -> c_7() , geq^#(n__0(), n__s(Y)) -> c_8() , geq^#(n__s(X), n__s(Y)) -> c_9(geq^#(activate(X), activate(Y))) , div^#(0(), n__s(Y)) -> c_10(0^#()) , div^#(s(X), n__s(Y)) -> c_11(if^#(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0())) , if^#(true(), X, Y) -> c_13(activate^#(X)) , if^#(false(), X, Y) -> c_14(activate^#(Y)) } Strict Trs: { 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(X) , 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(0(), n__s(Y)) -> 0() , div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(div(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 {3,8,9} by applications of Pre({3,8,9}) = {1,4,5,7,10,11}. Here rules are labeled as follows: DPs: { 1: minus^#(n__0(), Y) -> c_1(0^#()) , 2: minus^#(n__s(X), n__s(Y)) -> c_2(minus^#(activate(X), activate(Y))) , 3: 0^#() -> c_3() , 4: activate^#(X) -> c_4(X) , 5: activate^#(n__0()) -> c_5(0^#()) , 6: activate^#(n__s(X)) -> c_6(s^#(X)) , 7: s^#(X) -> c_12(X) , 8: geq^#(X, n__0()) -> c_7() , 9: geq^#(n__0(), n__s(Y)) -> c_8() , 10: geq^#(n__s(X), n__s(Y)) -> c_9(geq^#(activate(X), activate(Y))) , 11: div^#(0(), n__s(Y)) -> c_10(0^#()) , 12: div^#(s(X), n__s(Y)) -> c_11(if^#(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0())) , 13: if^#(true(), X, Y) -> c_13(activate^#(X)) , 14: if^#(false(), X, Y) -> c_14(activate^#(Y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(n__0(), Y) -> c_1(0^#()) , minus^#(n__s(X), n__s(Y)) -> c_2(minus^#(activate(X), activate(Y))) , activate^#(X) -> c_4(X) , activate^#(n__0()) -> c_5(0^#()) , activate^#(n__s(X)) -> c_6(s^#(X)) , s^#(X) -> c_12(X) , geq^#(n__s(X), n__s(Y)) -> c_9(geq^#(activate(X), activate(Y))) , div^#(0(), n__s(Y)) -> c_10(0^#()) , div^#(s(X), n__s(Y)) -> c_11(if^#(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0())) , if^#(true(), X, Y) -> c_13(activate^#(X)) , if^#(false(), X, Y) -> c_14(activate^#(Y)) } Strict Trs: { 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(X) , 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(0(), n__s(Y)) -> 0() , div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(div(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_3() , geq^#(X, n__0()) -> c_7() , geq^#(n__0(), n__s(Y)) -> c_8() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,4,8} by applications of Pre({1,4,8}) = {2,3,6,10,11}. Here rules are labeled as follows: DPs: { 1: minus^#(n__0(), Y) -> c_1(0^#()) , 2: minus^#(n__s(X), n__s(Y)) -> c_2(minus^#(activate(X), activate(Y))) , 3: activate^#(X) -> c_4(X) , 4: activate^#(n__0()) -> c_5(0^#()) , 5: activate^#(n__s(X)) -> c_6(s^#(X)) , 6: s^#(X) -> c_12(X) , 7: geq^#(n__s(X), n__s(Y)) -> c_9(geq^#(activate(X), activate(Y))) , 8: div^#(0(), n__s(Y)) -> c_10(0^#()) , 9: div^#(s(X), n__s(Y)) -> c_11(if^#(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0())) , 10: if^#(true(), X, Y) -> c_13(activate^#(X)) , 11: if^#(false(), X, Y) -> c_14(activate^#(Y)) , 12: 0^#() -> c_3() , 13: geq^#(X, n__0()) -> c_7() , 14: geq^#(n__0(), n__s(Y)) -> c_8() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(n__s(X), n__s(Y)) -> c_2(minus^#(activate(X), activate(Y))) , activate^#(X) -> c_4(X) , activate^#(n__s(X)) -> c_6(s^#(X)) , s^#(X) -> c_12(X) , geq^#(n__s(X), n__s(Y)) -> c_9(geq^#(activate(X), activate(Y))) , div^#(s(X), n__s(Y)) -> c_11(if^#(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0())) , if^#(true(), X, Y) -> c_13(activate^#(X)) , if^#(false(), X, Y) -> c_14(activate^#(Y)) } Strict Trs: { 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(X) , 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(0(), n__s(Y)) -> 0() , div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(div(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_1(0^#()) , 0^#() -> c_3() , activate^#(n__0()) -> c_5(0^#()) , geq^#(X, n__0()) -> c_7() , geq^#(n__0(), n__s(Y)) -> c_8() , div^#(0(), n__s(Y)) -> c_10(0^#()) } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..