MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { fact(X) -> if(zero(X), n__s(n__0()), n__prod(X, n__fact(n__p(X)))) , fact(X) -> n__fact(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) , zero(0()) -> true() , zero(s(X)) -> false() , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , 0() -> n__0() , s(X) -> n__s(X) , prod(X1, X2) -> n__prod(X1, X2) , prod(0(), X) -> 0() , prod(s(X), Y) -> add(Y, prod(X, Y)) , activate(X) -> X , activate(n__s(X)) -> s(activate(X)) , activate(n__0()) -> 0() , activate(n__prod(X1, X2)) -> prod(activate(X1), activate(X2)) , activate(n__fact(X)) -> fact(activate(X)) , activate(n__p(X)) -> p(activate(X)) , p(X) -> n__p(X) , p(s(X)) -> X } 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: { fact^#(X) -> c_1(if^#(zero(X), n__s(n__0()), n__prod(X, n__fact(n__p(X))))) , fact^#(X) -> c_2(X) , if^#(true(), X, Y) -> c_3(activate^#(X)) , if^#(false(), X, Y) -> c_4(activate^#(Y)) , activate^#(X) -> c_14(X) , activate^#(n__s(X)) -> c_15(s^#(activate(X))) , activate^#(n__0()) -> c_16(0^#()) , activate^#(n__prod(X1, X2)) -> c_17(prod^#(activate(X1), activate(X2))) , activate^#(n__fact(X)) -> c_18(fact^#(activate(X))) , activate^#(n__p(X)) -> c_19(p^#(activate(X))) , zero^#(0()) -> c_5() , zero^#(s(X)) -> c_6() , add^#(0(), X) -> c_7(X) , add^#(s(X), Y) -> c_8(s^#(add(X, Y))) , s^#(X) -> c_10(X) , 0^#() -> c_9() , prod^#(X1, X2) -> c_11(X1, X2) , prod^#(0(), X) -> c_12(0^#()) , prod^#(s(X), Y) -> c_13(add^#(Y, prod(X, Y))) , p^#(X) -> c_20(X) , p^#(s(X)) -> c_21(X) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { fact^#(X) -> c_1(if^#(zero(X), n__s(n__0()), n__prod(X, n__fact(n__p(X))))) , fact^#(X) -> c_2(X) , if^#(true(), X, Y) -> c_3(activate^#(X)) , if^#(false(), X, Y) -> c_4(activate^#(Y)) , activate^#(X) -> c_14(X) , activate^#(n__s(X)) -> c_15(s^#(activate(X))) , activate^#(n__0()) -> c_16(0^#()) , activate^#(n__prod(X1, X2)) -> c_17(prod^#(activate(X1), activate(X2))) , activate^#(n__fact(X)) -> c_18(fact^#(activate(X))) , activate^#(n__p(X)) -> c_19(p^#(activate(X))) , zero^#(0()) -> c_5() , zero^#(s(X)) -> c_6() , add^#(0(), X) -> c_7(X) , add^#(s(X), Y) -> c_8(s^#(add(X, Y))) , s^#(X) -> c_10(X) , 0^#() -> c_9() , prod^#(X1, X2) -> c_11(X1, X2) , prod^#(0(), X) -> c_12(0^#()) , prod^#(s(X), Y) -> c_13(add^#(Y, prod(X, Y))) , p^#(X) -> c_20(X) , p^#(s(X)) -> c_21(X) } Strict Trs: { fact(X) -> if(zero(X), n__s(n__0()), n__prod(X, n__fact(n__p(X)))) , fact(X) -> n__fact(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) , zero(0()) -> true() , zero(s(X)) -> false() , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , 0() -> n__0() , s(X) -> n__s(X) , prod(X1, X2) -> n__prod(X1, X2) , prod(0(), X) -> 0() , prod(s(X), Y) -> add(Y, prod(X, Y)) , activate(X) -> X , activate(n__s(X)) -> s(activate(X)) , activate(n__0()) -> 0() , activate(n__prod(X1, X2)) -> prod(activate(X1), activate(X2)) , activate(n__fact(X)) -> fact(activate(X)) , activate(n__p(X)) -> p(activate(X)) , p(X) -> n__p(X) , p(s(X)) -> X } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {11,12,16} by applications of Pre({11,12,16}) = {2,5,7,13,15,17,18,20,21}. Here rules are labeled as follows: DPs: { 1: fact^#(X) -> c_1(if^#(zero(X), n__s(n__0()), n__prod(X, n__fact(n__p(X))))) , 2: fact^#(X) -> c_2(X) , 3: if^#(true(), X, Y) -> c_3(activate^#(X)) , 4: if^#(false(), X, Y) -> c_4(activate^#(Y)) , 5: activate^#(X) -> c_14(X) , 6: activate^#(n__s(X)) -> c_15(s^#(activate(X))) , 7: activate^#(n__0()) -> c_16(0^#()) , 8: activate^#(n__prod(X1, X2)) -> c_17(prod^#(activate(X1), activate(X2))) , 9: activate^#(n__fact(X)) -> c_18(fact^#(activate(X))) , 10: activate^#(n__p(X)) -> c_19(p^#(activate(X))) , 11: zero^#(0()) -> c_5() , 12: zero^#(s(X)) -> c_6() , 13: add^#(0(), X) -> c_7(X) , 14: add^#(s(X), Y) -> c_8(s^#(add(X, Y))) , 15: s^#(X) -> c_10(X) , 16: 0^#() -> c_9() , 17: prod^#(X1, X2) -> c_11(X1, X2) , 18: prod^#(0(), X) -> c_12(0^#()) , 19: prod^#(s(X), Y) -> c_13(add^#(Y, prod(X, Y))) , 20: p^#(X) -> c_20(X) , 21: p^#(s(X)) -> c_21(X) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { fact^#(X) -> c_1(if^#(zero(X), n__s(n__0()), n__prod(X, n__fact(n__p(X))))) , fact^#(X) -> c_2(X) , if^#(true(), X, Y) -> c_3(activate^#(X)) , if^#(false(), X, Y) -> c_4(activate^#(Y)) , activate^#(X) -> c_14(X) , activate^#(n__s(X)) -> c_15(s^#(activate(X))) , activate^#(n__0()) -> c_16(0^#()) , activate^#(n__prod(X1, X2)) -> c_17(prod^#(activate(X1), activate(X2))) , activate^#(n__fact(X)) -> c_18(fact^#(activate(X))) , activate^#(n__p(X)) -> c_19(p^#(activate(X))) , add^#(0(), X) -> c_7(X) , add^#(s(X), Y) -> c_8(s^#(add(X, Y))) , s^#(X) -> c_10(X) , prod^#(X1, X2) -> c_11(X1, X2) , prod^#(0(), X) -> c_12(0^#()) , prod^#(s(X), Y) -> c_13(add^#(Y, prod(X, Y))) , p^#(X) -> c_20(X) , p^#(s(X)) -> c_21(X) } Strict Trs: { fact(X) -> if(zero(X), n__s(n__0()), n__prod(X, n__fact(n__p(X)))) , fact(X) -> n__fact(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) , zero(0()) -> true() , zero(s(X)) -> false() , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , 0() -> n__0() , s(X) -> n__s(X) , prod(X1, X2) -> n__prod(X1, X2) , prod(0(), X) -> 0() , prod(s(X), Y) -> add(Y, prod(X, Y)) , activate(X) -> X , activate(n__s(X)) -> s(activate(X)) , activate(n__0()) -> 0() , activate(n__prod(X1, X2)) -> prod(activate(X1), activate(X2)) , activate(n__fact(X)) -> fact(activate(X)) , activate(n__p(X)) -> p(activate(X)) , p(X) -> n__p(X) , p(s(X)) -> X } Weak DPs: { zero^#(0()) -> c_5() , zero^#(s(X)) -> c_6() , 0^#() -> c_9() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {7,15} by applications of Pre({7,15}) = {2,3,4,5,8,11,13,14,17,18}. Here rules are labeled as follows: DPs: { 1: fact^#(X) -> c_1(if^#(zero(X), n__s(n__0()), n__prod(X, n__fact(n__p(X))))) , 2: fact^#(X) -> c_2(X) , 3: if^#(true(), X, Y) -> c_3(activate^#(X)) , 4: if^#(false(), X, Y) -> c_4(activate^#(Y)) , 5: activate^#(X) -> c_14(X) , 6: activate^#(n__s(X)) -> c_15(s^#(activate(X))) , 7: activate^#(n__0()) -> c_16(0^#()) , 8: activate^#(n__prod(X1, X2)) -> c_17(prod^#(activate(X1), activate(X2))) , 9: activate^#(n__fact(X)) -> c_18(fact^#(activate(X))) , 10: activate^#(n__p(X)) -> c_19(p^#(activate(X))) , 11: add^#(0(), X) -> c_7(X) , 12: add^#(s(X), Y) -> c_8(s^#(add(X, Y))) , 13: s^#(X) -> c_10(X) , 14: prod^#(X1, X2) -> c_11(X1, X2) , 15: prod^#(0(), X) -> c_12(0^#()) , 16: prod^#(s(X), Y) -> c_13(add^#(Y, prod(X, Y))) , 17: p^#(X) -> c_20(X) , 18: p^#(s(X)) -> c_21(X) , 19: zero^#(0()) -> c_5() , 20: zero^#(s(X)) -> c_6() , 21: 0^#() -> c_9() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { fact^#(X) -> c_1(if^#(zero(X), n__s(n__0()), n__prod(X, n__fact(n__p(X))))) , fact^#(X) -> c_2(X) , if^#(true(), X, Y) -> c_3(activate^#(X)) , if^#(false(), X, Y) -> c_4(activate^#(Y)) , activate^#(X) -> c_14(X) , activate^#(n__s(X)) -> c_15(s^#(activate(X))) , activate^#(n__prod(X1, X2)) -> c_17(prod^#(activate(X1), activate(X2))) , activate^#(n__fact(X)) -> c_18(fact^#(activate(X))) , activate^#(n__p(X)) -> c_19(p^#(activate(X))) , add^#(0(), X) -> c_7(X) , add^#(s(X), Y) -> c_8(s^#(add(X, Y))) , s^#(X) -> c_10(X) , prod^#(X1, X2) -> c_11(X1, X2) , prod^#(s(X), Y) -> c_13(add^#(Y, prod(X, Y))) , p^#(X) -> c_20(X) , p^#(s(X)) -> c_21(X) } Strict Trs: { fact(X) -> if(zero(X), n__s(n__0()), n__prod(X, n__fact(n__p(X)))) , fact(X) -> n__fact(X) , if(true(), X, Y) -> activate(X) , if(false(), X, Y) -> activate(Y) , zero(0()) -> true() , zero(s(X)) -> false() , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , 0() -> n__0() , s(X) -> n__s(X) , prod(X1, X2) -> n__prod(X1, X2) , prod(0(), X) -> 0() , prod(s(X), Y) -> add(Y, prod(X, Y)) , activate(X) -> X , activate(n__s(X)) -> s(activate(X)) , activate(n__0()) -> 0() , activate(n__prod(X1, X2)) -> prod(activate(X1), activate(X2)) , activate(n__fact(X)) -> fact(activate(X)) , activate(n__p(X)) -> p(activate(X)) , p(X) -> n__p(X) , p(s(X)) -> X } Weak DPs: { activate^#(n__0()) -> c_16(0^#()) , zero^#(0()) -> c_5() , zero^#(s(X)) -> c_6() , 0^#() -> c_9() , prod^#(0(), X) -> c_12(0^#()) } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..