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