MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), k(n(s(x), s(y)), s(s(0()))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y)))) , if(true(), x, y) -> x , if(false(), x, y) -> y , and(x, false()) -> false() , and(true(), true()) -> true() , f(s(x)) -> h(x) , f(0()) -> true() , t(x) -> p(x, x) , k(s(x), s(y)) -> s(k(minus(x, y), s(y))) , k(0(), s(y)) -> 0() , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) , m(x, 0()) -> x , m(s(x), s(y)) -> s(m(x, y)) , m(0(), y) -> y , n(x, 0()) -> 0() , n(s(x), s(y)) -> s(n(x, y)) , n(0(), y) -> 0() , p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x)) , p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) , p(0(), y) -> y , p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y)))) , gt(s(x), s(y)) -> gt(x, y) , gt(s(x), 0()) -> true() , gt(0(), y) -> false() , not(x) -> if(x, false(), true()) , id(x) -> x , h(s(x)) -> f(x) , h(0()) -> false() } 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: { g^#(s(x), s(y)) -> c_1(if^#(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), k(n(s(x), s(y)), s(s(0()))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))) , if^#(true(), x, y) -> c_2(x) , if^#(false(), x, y) -> c_3(y) , and^#(x, false()) -> c_4() , and^#(true(), true()) -> c_5() , f^#(s(x)) -> c_6(h^#(x)) , f^#(0()) -> c_7() , h^#(s(x)) -> c_28(f^#(x)) , h^#(0()) -> c_29() , t^#(x) -> c_8(p^#(x, x)) , p^#(s(x), x) -> c_19(p^#(if(gt(x, x), id(x), id(x)), s(x))) , p^#(s(x), s(y)) -> c_20(p^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))) , p^#(0(), y) -> c_21(y) , p^#(id(x), s(y)) -> c_22(p^#(x, if(gt(s(y), y), y, s(y)))) , k^#(s(x), s(y)) -> c_9(k^#(minus(x, y), s(y))) , k^#(0(), s(y)) -> c_10() , minus^#(x, 0()) -> c_11(x) , minus^#(s(x), s(y)) -> c_12(minus^#(x, y)) , m^#(x, 0()) -> c_13(x) , m^#(s(x), s(y)) -> c_14(m^#(x, y)) , m^#(0(), y) -> c_15(y) , n^#(x, 0()) -> c_16() , n^#(s(x), s(y)) -> c_17(n^#(x, y)) , n^#(0(), y) -> c_18() , gt^#(s(x), s(y)) -> c_23(gt^#(x, y)) , gt^#(s(x), 0()) -> c_24() , gt^#(0(), y) -> c_25() , not^#(x) -> c_26(if^#(x, false(), true())) , id^#(x) -> c_27(x) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { g^#(s(x), s(y)) -> c_1(if^#(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), k(n(s(x), s(y)), s(s(0()))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))) , if^#(true(), x, y) -> c_2(x) , if^#(false(), x, y) -> c_3(y) , and^#(x, false()) -> c_4() , and^#(true(), true()) -> c_5() , f^#(s(x)) -> c_6(h^#(x)) , f^#(0()) -> c_7() , h^#(s(x)) -> c_28(f^#(x)) , h^#(0()) -> c_29() , t^#(x) -> c_8(p^#(x, x)) , p^#(s(x), x) -> c_19(p^#(if(gt(x, x), id(x), id(x)), s(x))) , p^#(s(x), s(y)) -> c_20(p^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))) , p^#(0(), y) -> c_21(y) , p^#(id(x), s(y)) -> c_22(p^#(x, if(gt(s(y), y), y, s(y)))) , k^#(s(x), s(y)) -> c_9(k^#(minus(x, y), s(y))) , k^#(0(), s(y)) -> c_10() , minus^#(x, 0()) -> c_11(x) , minus^#(s(x), s(y)) -> c_12(minus^#(x, y)) , m^#(x, 0()) -> c_13(x) , m^#(s(x), s(y)) -> c_14(m^#(x, y)) , m^#(0(), y) -> c_15(y) , n^#(x, 0()) -> c_16() , n^#(s(x), s(y)) -> c_17(n^#(x, y)) , n^#(0(), y) -> c_18() , gt^#(s(x), s(y)) -> c_23(gt^#(x, y)) , gt^#(s(x), 0()) -> c_24() , gt^#(0(), y) -> c_25() , not^#(x) -> c_26(if^#(x, false(), true())) , id^#(x) -> c_27(x) } Strict Trs: { g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), k(n(s(x), s(y)), s(s(0()))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y)))) , if(true(), x, y) -> x , if(false(), x, y) -> y , and(x, false()) -> false() , and(true(), true()) -> true() , f(s(x)) -> h(x) , f(0()) -> true() , t(x) -> p(x, x) , k(s(x), s(y)) -> s(k(minus(x, y), s(y))) , k(0(), s(y)) -> 0() , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) , m(x, 0()) -> x , m(s(x), s(y)) -> s(m(x, y)) , m(0(), y) -> y , n(x, 0()) -> 0() , n(s(x), s(y)) -> s(n(x, y)) , n(0(), y) -> 0() , p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x)) , p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) , p(0(), y) -> y , p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y)))) , gt(s(x), s(y)) -> gt(x, y) , gt(s(x), 0()) -> true() , gt(0(), y) -> false() , not(x) -> if(x, false(), true()) , id(x) -> x , h(s(x)) -> f(x) , h(0()) -> false() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {4,5,7,9,16,22,24,26,27} by applications of Pre({4,5,7,9,16,22,24,26,27}) = {2,3,6,8,13,15,17,19,21,23,25,29}. Here rules are labeled as follows: DPs: { 1: g^#(s(x), s(y)) -> c_1(if^#(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), k(n(s(x), s(y)), s(s(0()))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))) , 2: if^#(true(), x, y) -> c_2(x) , 3: if^#(false(), x, y) -> c_3(y) , 4: and^#(x, false()) -> c_4() , 5: and^#(true(), true()) -> c_5() , 6: f^#(s(x)) -> c_6(h^#(x)) , 7: f^#(0()) -> c_7() , 8: h^#(s(x)) -> c_28(f^#(x)) , 9: h^#(0()) -> c_29() , 10: t^#(x) -> c_8(p^#(x, x)) , 11: p^#(s(x), x) -> c_19(p^#(if(gt(x, x), id(x), id(x)), s(x))) , 12: p^#(s(x), s(y)) -> c_20(p^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))) , 13: p^#(0(), y) -> c_21(y) , 14: p^#(id(x), s(y)) -> c_22(p^#(x, if(gt(s(y), y), y, s(y)))) , 15: k^#(s(x), s(y)) -> c_9(k^#(minus(x, y), s(y))) , 16: k^#(0(), s(y)) -> c_10() , 17: minus^#(x, 0()) -> c_11(x) , 18: minus^#(s(x), s(y)) -> c_12(minus^#(x, y)) , 19: m^#(x, 0()) -> c_13(x) , 20: m^#(s(x), s(y)) -> c_14(m^#(x, y)) , 21: m^#(0(), y) -> c_15(y) , 22: n^#(x, 0()) -> c_16() , 23: n^#(s(x), s(y)) -> c_17(n^#(x, y)) , 24: n^#(0(), y) -> c_18() , 25: gt^#(s(x), s(y)) -> c_23(gt^#(x, y)) , 26: gt^#(s(x), 0()) -> c_24() , 27: gt^#(0(), y) -> c_25() , 28: not^#(x) -> c_26(if^#(x, false(), true())) , 29: id^#(x) -> c_27(x) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { g^#(s(x), s(y)) -> c_1(if^#(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), k(n(s(x), s(y)), s(s(0()))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y))))) , if^#(true(), x, y) -> c_2(x) , if^#(false(), x, y) -> c_3(y) , f^#(s(x)) -> c_6(h^#(x)) , h^#(s(x)) -> c_28(f^#(x)) , t^#(x) -> c_8(p^#(x, x)) , p^#(s(x), x) -> c_19(p^#(if(gt(x, x), id(x), id(x)), s(x))) , p^#(s(x), s(y)) -> c_20(p^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))) , p^#(0(), y) -> c_21(y) , p^#(id(x), s(y)) -> c_22(p^#(x, if(gt(s(y), y), y, s(y)))) , k^#(s(x), s(y)) -> c_9(k^#(minus(x, y), s(y))) , minus^#(x, 0()) -> c_11(x) , minus^#(s(x), s(y)) -> c_12(minus^#(x, y)) , m^#(x, 0()) -> c_13(x) , m^#(s(x), s(y)) -> c_14(m^#(x, y)) , m^#(0(), y) -> c_15(y) , n^#(s(x), s(y)) -> c_17(n^#(x, y)) , gt^#(s(x), s(y)) -> c_23(gt^#(x, y)) , not^#(x) -> c_26(if^#(x, false(), true())) , id^#(x) -> c_27(x) } Strict Trs: { g(s(x), s(y)) -> if(and(f(s(x)), f(s(y))), t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), k(n(s(x), s(y)), s(s(0()))))), g(minus(m(x, y), n(x, y)), n(s(x), s(y)))) , if(true(), x, y) -> x , if(false(), x, y) -> y , and(x, false()) -> false() , and(true(), true()) -> true() , f(s(x)) -> h(x) , f(0()) -> true() , t(x) -> p(x, x) , k(s(x), s(y)) -> s(k(minus(x, y), s(y))) , k(0(), s(y)) -> 0() , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) , m(x, 0()) -> x , m(s(x), s(y)) -> s(m(x, y)) , m(0(), y) -> y , n(x, 0()) -> 0() , n(s(x), s(y)) -> s(n(x, y)) , n(0(), y) -> 0() , p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x)) , p(s(x), s(y)) -> s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) , p(0(), y) -> y , p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y)))) , gt(s(x), s(y)) -> gt(x, y) , gt(s(x), 0()) -> true() , gt(0(), y) -> false() , not(x) -> if(x, false(), true()) , id(x) -> x , h(s(x)) -> f(x) , h(0()) -> false() } Weak DPs: { and^#(x, false()) -> c_4() , and^#(true(), true()) -> c_5() , f^#(0()) -> c_7() , h^#(0()) -> c_29() , k^#(0(), s(y)) -> c_10() , n^#(x, 0()) -> c_16() , n^#(0(), y) -> c_18() , gt^#(s(x), 0()) -> c_24() , gt^#(0(), y) -> c_25() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..