MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a__fact(X) -> a__if(a__zero(mark(X)), s(0()), prod(X, fact(p(X)))) , a__fact(X) -> fact(X) , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , a__zero(X) -> zero(X) , a__zero(s(X)) -> false() , a__zero(0()) -> true() , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) , mark(fact(X)) -> a__fact(mark(X)) , mark(p(X)) -> a__p(mark(X)) , mark(true()) -> true() , mark(false()) -> false() , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , mark(zero(X)) -> a__zero(mark(X)) , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) , a__add(X1, X2) -> add(X1, X2) , a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) , a__add(0(), X) -> mark(X) , a__prod(X1, X2) -> prod(X1, X2) , a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) , a__prod(0(), X) -> 0() , a__p(X) -> p(X) , a__p(s(X)) -> mark(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) '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 minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) '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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { a__fact^#(X) -> c_1(a__if^#(a__zero(mark(X)), s(0()), prod(X, fact(p(X))))) , a__fact^#(X) -> c_2(X) , a__if^#(X1, X2, X3) -> c_3(X1, X2, X3) , a__if^#(true(), X, Y) -> c_4(mark^#(X)) , a__if^#(false(), X, Y) -> c_5(mark^#(Y)) , mark^#(s(X)) -> c_9(mark^#(X)) , mark^#(0()) -> c_10() , mark^#(prod(X1, X2)) -> c_11(a__prod^#(mark(X1), mark(X2))) , mark^#(fact(X)) -> c_12(a__fact^#(mark(X))) , mark^#(p(X)) -> c_13(a__p^#(mark(X))) , mark^#(true()) -> c_14() , mark^#(false()) -> c_15() , mark^#(if(X1, X2, X3)) -> c_16(a__if^#(mark(X1), X2, X3)) , mark^#(zero(X)) -> c_17(a__zero^#(mark(X))) , mark^#(add(X1, X2)) -> c_18(a__add^#(mark(X1), mark(X2))) , a__zero^#(X) -> c_6(X) , a__zero^#(s(X)) -> c_7() , a__zero^#(0()) -> c_8() , a__prod^#(X1, X2) -> c_22(X1, X2) , a__prod^#(s(X), Y) -> c_23(a__add^#(mark(Y), a__prod(mark(X), mark(Y)))) , a__prod^#(0(), X) -> c_24() , a__p^#(X) -> c_25(X) , a__p^#(s(X)) -> c_26(mark^#(X)) , a__add^#(X1, X2) -> c_19(X1, X2) , a__add^#(s(X), Y) -> c_20(a__add^#(mark(X), mark(Y))) , a__add^#(0(), X) -> c_21(mark^#(X)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__fact^#(X) -> c_1(a__if^#(a__zero(mark(X)), s(0()), prod(X, fact(p(X))))) , a__fact^#(X) -> c_2(X) , a__if^#(X1, X2, X3) -> c_3(X1, X2, X3) , a__if^#(true(), X, Y) -> c_4(mark^#(X)) , a__if^#(false(), X, Y) -> c_5(mark^#(Y)) , mark^#(s(X)) -> c_9(mark^#(X)) , mark^#(0()) -> c_10() , mark^#(prod(X1, X2)) -> c_11(a__prod^#(mark(X1), mark(X2))) , mark^#(fact(X)) -> c_12(a__fact^#(mark(X))) , mark^#(p(X)) -> c_13(a__p^#(mark(X))) , mark^#(true()) -> c_14() , mark^#(false()) -> c_15() , mark^#(if(X1, X2, X3)) -> c_16(a__if^#(mark(X1), X2, X3)) , mark^#(zero(X)) -> c_17(a__zero^#(mark(X))) , mark^#(add(X1, X2)) -> c_18(a__add^#(mark(X1), mark(X2))) , a__zero^#(X) -> c_6(X) , a__zero^#(s(X)) -> c_7() , a__zero^#(0()) -> c_8() , a__prod^#(X1, X2) -> c_22(X1, X2) , a__prod^#(s(X), Y) -> c_23(a__add^#(mark(Y), a__prod(mark(X), mark(Y)))) , a__prod^#(0(), X) -> c_24() , a__p^#(X) -> c_25(X) , a__p^#(s(X)) -> c_26(mark^#(X)) , a__add^#(X1, X2) -> c_19(X1, X2) , a__add^#(s(X), Y) -> c_20(a__add^#(mark(X), mark(Y))) , a__add^#(0(), X) -> c_21(mark^#(X)) } Strict Trs: { a__fact(X) -> a__if(a__zero(mark(X)), s(0()), prod(X, fact(p(X)))) , a__fact(X) -> fact(X) , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , a__zero(X) -> zero(X) , a__zero(s(X)) -> false() , a__zero(0()) -> true() , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) , mark(fact(X)) -> a__fact(mark(X)) , mark(p(X)) -> a__p(mark(X)) , mark(true()) -> true() , mark(false()) -> false() , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , mark(zero(X)) -> a__zero(mark(X)) , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) , a__add(X1, X2) -> add(X1, X2) , a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) , a__add(0(), X) -> mark(X) , a__prod(X1, X2) -> prod(X1, X2) , a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) , a__prod(0(), X) -> 0() , a__p(X) -> p(X) , a__p(s(X)) -> mark(X) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {7,11,12,17,18,21} by applications of Pre({7,11,12,17,18,21}) = {2,3,4,5,6,8,14,16,19,22,23,24,26}. Here rules are labeled as follows: DPs: { 1: a__fact^#(X) -> c_1(a__if^#(a__zero(mark(X)), s(0()), prod(X, fact(p(X))))) , 2: a__fact^#(X) -> c_2(X) , 3: a__if^#(X1, X2, X3) -> c_3(X1, X2, X3) , 4: a__if^#(true(), X, Y) -> c_4(mark^#(X)) , 5: a__if^#(false(), X, Y) -> c_5(mark^#(Y)) , 6: mark^#(s(X)) -> c_9(mark^#(X)) , 7: mark^#(0()) -> c_10() , 8: mark^#(prod(X1, X2)) -> c_11(a__prod^#(mark(X1), mark(X2))) , 9: mark^#(fact(X)) -> c_12(a__fact^#(mark(X))) , 10: mark^#(p(X)) -> c_13(a__p^#(mark(X))) , 11: mark^#(true()) -> c_14() , 12: mark^#(false()) -> c_15() , 13: mark^#(if(X1, X2, X3)) -> c_16(a__if^#(mark(X1), X2, X3)) , 14: mark^#(zero(X)) -> c_17(a__zero^#(mark(X))) , 15: mark^#(add(X1, X2)) -> c_18(a__add^#(mark(X1), mark(X2))) , 16: a__zero^#(X) -> c_6(X) , 17: a__zero^#(s(X)) -> c_7() , 18: a__zero^#(0()) -> c_8() , 19: a__prod^#(X1, X2) -> c_22(X1, X2) , 20: a__prod^#(s(X), Y) -> c_23(a__add^#(mark(Y), a__prod(mark(X), mark(Y)))) , 21: a__prod^#(0(), X) -> c_24() , 22: a__p^#(X) -> c_25(X) , 23: a__p^#(s(X)) -> c_26(mark^#(X)) , 24: a__add^#(X1, X2) -> c_19(X1, X2) , 25: a__add^#(s(X), Y) -> c_20(a__add^#(mark(X), mark(Y))) , 26: a__add^#(0(), X) -> c_21(mark^#(X)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__fact^#(X) -> c_1(a__if^#(a__zero(mark(X)), s(0()), prod(X, fact(p(X))))) , a__fact^#(X) -> c_2(X) , a__if^#(X1, X2, X3) -> c_3(X1, X2, X3) , a__if^#(true(), X, Y) -> c_4(mark^#(X)) , a__if^#(false(), X, Y) -> c_5(mark^#(Y)) , mark^#(s(X)) -> c_9(mark^#(X)) , mark^#(prod(X1, X2)) -> c_11(a__prod^#(mark(X1), mark(X2))) , mark^#(fact(X)) -> c_12(a__fact^#(mark(X))) , mark^#(p(X)) -> c_13(a__p^#(mark(X))) , mark^#(if(X1, X2, X3)) -> c_16(a__if^#(mark(X1), X2, X3)) , mark^#(zero(X)) -> c_17(a__zero^#(mark(X))) , mark^#(add(X1, X2)) -> c_18(a__add^#(mark(X1), mark(X2))) , a__zero^#(X) -> c_6(X) , a__prod^#(X1, X2) -> c_22(X1, X2) , a__prod^#(s(X), Y) -> c_23(a__add^#(mark(Y), a__prod(mark(X), mark(Y)))) , a__p^#(X) -> c_25(X) , a__p^#(s(X)) -> c_26(mark^#(X)) , a__add^#(X1, X2) -> c_19(X1, X2) , a__add^#(s(X), Y) -> c_20(a__add^#(mark(X), mark(Y))) , a__add^#(0(), X) -> c_21(mark^#(X)) } Strict Trs: { a__fact(X) -> a__if(a__zero(mark(X)), s(0()), prod(X, fact(p(X)))) , a__fact(X) -> fact(X) , a__if(X1, X2, X3) -> if(X1, X2, X3) , a__if(true(), X, Y) -> mark(X) , a__if(false(), X, Y) -> mark(Y) , a__zero(X) -> zero(X) , a__zero(s(X)) -> false() , a__zero(0()) -> true() , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) , mark(fact(X)) -> a__fact(mark(X)) , mark(p(X)) -> a__p(mark(X)) , mark(true()) -> true() , mark(false()) -> false() , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) , mark(zero(X)) -> a__zero(mark(X)) , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) , a__add(X1, X2) -> add(X1, X2) , a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) , a__add(0(), X) -> mark(X) , a__prod(X1, X2) -> prod(X1, X2) , a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) , a__prod(0(), X) -> 0() , a__p(X) -> p(X) , a__p(s(X)) -> mark(X) } Weak DPs: { mark^#(0()) -> c_10() , mark^#(true()) -> c_14() , mark^#(false()) -> c_15() , a__zero^#(s(X)) -> c_7() , a__zero^#(0()) -> c_8() , a__prod^#(0(), X) -> c_24() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..