MAYBE 901.15/297.03 MAYBE 901.15/297.03 901.15/297.03 We are left with following problem, upon which TcT provides the 901.15/297.03 certificate MAYBE. 901.15/297.03 901.15/297.03 Strict Trs: 901.15/297.03 { g(s(x), s(y)) -> 901.15/297.03 if(and(f(s(x)), f(s(y))), 901.15/297.03 t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), 901.15/297.03 k(n(s(x), s(y)), s(s(0()))))), 901.15/297.03 g(minus(m(x, y), n(x, y)), n(s(x), s(y)))) 901.15/297.03 , if(true(), x, y) -> x 901.15/297.03 , if(false(), x, y) -> y 901.15/297.03 , and(x, false()) -> false() 901.15/297.03 , and(true(), true()) -> true() 901.15/297.03 , f(s(x)) -> h(x) 901.15/297.03 , f(0()) -> true() 901.15/297.03 , t(x) -> p(x, x) 901.15/297.03 , k(s(x), s(y)) -> s(k(minus(x, y), s(y))) 901.15/297.03 , k(0(), s(y)) -> 0() 901.15/297.03 , minus(x, 0()) -> x 901.15/297.03 , minus(s(x), s(y)) -> minus(x, y) 901.15/297.03 , m(x, 0()) -> x 901.15/297.03 , m(s(x), s(y)) -> s(m(x, y)) 901.15/297.03 , m(0(), y) -> y 901.15/297.03 , n(x, 0()) -> 0() 901.15/297.03 , n(s(x), s(y)) -> s(n(x, y)) 901.15/297.03 , n(0(), y) -> 0() 901.15/297.03 , p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x)) 901.15/297.03 , p(s(x), s(y)) -> 901.15/297.03 s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 901.15/297.03 , p(0(), y) -> y 901.15/297.03 , p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y)))) 901.15/297.03 , gt(s(x), s(y)) -> gt(x, y) 901.15/297.03 , gt(s(x), 0()) -> true() 901.15/297.03 , gt(0(), y) -> false() 901.15/297.03 , not(x) -> if(x, false(), true()) 901.15/297.03 , id(x) -> x 901.15/297.03 , h(s(x)) -> f(x) 901.15/297.03 , h(0()) -> false() } 901.15/297.03 Obligation: 901.15/297.03 runtime complexity 901.15/297.03 Answer: 901.15/297.03 MAYBE 901.15/297.03 901.15/297.03 None of the processors succeeded. 901.15/297.03 901.15/297.03 Details of failed attempt(s): 901.15/297.03 ----------------------------- 901.15/297.03 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 901.15/297.03 following reason: 901.15/297.03 901.15/297.03 Computation stopped due to timeout after 297.0 seconds. 901.15/297.03 901.15/297.03 2) 'Best' failed due to the following reason: 901.15/297.03 901.15/297.03 None of the processors succeeded. 901.15/297.03 901.15/297.03 Details of failed attempt(s): 901.15/297.03 ----------------------------- 901.15/297.03 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 901.15/297.03 seconds)' failed due to the following reason: 901.15/297.03 901.15/297.03 Computation stopped due to timeout after 148.0 seconds. 901.15/297.03 901.15/297.03 2) 'Best' failed due to the following reason: 901.15/297.03 901.15/297.03 None of the processors succeeded. 901.15/297.03 901.15/297.03 Details of failed attempt(s): 901.15/297.03 ----------------------------- 901.15/297.03 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 901.15/297.03 following reason: 901.15/297.03 901.15/297.03 The processor is inapplicable, reason: 901.15/297.03 Processor only applicable for innermost runtime complexity analysis 901.15/297.03 901.15/297.03 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 901.15/297.03 to the following reason: 901.15/297.03 901.15/297.03 The processor is inapplicable, reason: 901.15/297.03 Processor only applicable for innermost runtime complexity analysis 901.15/297.03 901.15/297.03 901.15/297.03 3) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 901.15/297.03 failed due to the following reason: 901.15/297.03 901.15/297.03 None of the processors succeeded. 901.15/297.03 901.15/297.03 Details of failed attempt(s): 901.15/297.03 ----------------------------- 901.15/297.03 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 901.15/297.03 failed due to the following reason: 901.15/297.03 901.15/297.03 match-boundness of the problem could not be verified. 901.15/297.03 901.15/297.03 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 901.15/297.03 failed due to the following reason: 901.15/297.03 901.15/297.03 match-boundness of the problem could not be verified. 901.15/297.03 901.15/297.03 901.15/297.03 901.15/297.03 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 901.15/297.03 the following reason: 901.15/297.03 901.15/297.03 We add the following weak dependency pairs: 901.15/297.03 901.15/297.03 Strict DPs: 901.15/297.03 { g^#(s(x), s(y)) -> 901.15/297.03 c_1(if^#(and(f(s(x)), f(s(y))), 901.15/297.03 t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), 901.15/297.03 k(n(s(x), s(y)), s(s(0()))))), 901.15/297.03 g(minus(m(x, y), n(x, y)), n(s(x), s(y))))) 901.15/297.03 , if^#(true(), x, y) -> c_2(x) 901.15/297.03 , if^#(false(), x, y) -> c_3(y) 901.15/297.03 , and^#(x, false()) -> c_4() 901.15/297.03 , and^#(true(), true()) -> c_5() 901.15/297.03 , f^#(s(x)) -> c_6(h^#(x)) 901.15/297.03 , f^#(0()) -> c_7() 901.15/297.03 , h^#(s(x)) -> c_28(f^#(x)) 901.15/297.03 , h^#(0()) -> c_29() 901.15/297.03 , t^#(x) -> c_8(p^#(x, x)) 901.15/297.03 , p^#(s(x), x) -> c_19(p^#(if(gt(x, x), id(x), id(x)), s(x))) 901.15/297.03 , p^#(s(x), s(y)) -> 901.15/297.03 c_20(p^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))) 901.15/297.03 , p^#(0(), y) -> c_21(y) 901.15/297.03 , p^#(id(x), s(y)) -> c_22(p^#(x, if(gt(s(y), y), y, s(y)))) 901.15/297.03 , k^#(s(x), s(y)) -> c_9(k^#(minus(x, y), s(y))) 901.15/297.03 , k^#(0(), s(y)) -> c_10() 901.15/297.03 , minus^#(x, 0()) -> c_11(x) 901.15/297.03 , minus^#(s(x), s(y)) -> c_12(minus^#(x, y)) 901.15/297.03 , m^#(x, 0()) -> c_13(x) 901.15/297.03 , m^#(s(x), s(y)) -> c_14(m^#(x, y)) 901.15/297.03 , m^#(0(), y) -> c_15(y) 901.15/297.03 , n^#(x, 0()) -> c_16() 901.15/297.03 , n^#(s(x), s(y)) -> c_17(n^#(x, y)) 901.15/297.03 , n^#(0(), y) -> c_18() 901.15/297.03 , gt^#(s(x), s(y)) -> c_23(gt^#(x, y)) 901.15/297.03 , gt^#(s(x), 0()) -> c_24() 901.15/297.03 , gt^#(0(), y) -> c_25() 901.15/297.03 , not^#(x) -> c_26(if^#(x, false(), true())) 901.15/297.03 , id^#(x) -> c_27(x) } 901.15/297.03 901.15/297.03 and mark the set of starting terms. 901.15/297.03 901.15/297.03 We are left with following problem, upon which TcT provides the 901.15/297.03 certificate MAYBE. 901.15/297.03 901.15/297.03 Strict DPs: 901.15/297.03 { g^#(s(x), s(y)) -> 901.15/297.03 c_1(if^#(and(f(s(x)), f(s(y))), 901.15/297.03 t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), 901.15/297.03 k(n(s(x), s(y)), s(s(0()))))), 901.15/297.03 g(minus(m(x, y), n(x, y)), n(s(x), s(y))))) 901.15/297.03 , if^#(true(), x, y) -> c_2(x) 901.15/297.03 , if^#(false(), x, y) -> c_3(y) 901.15/297.03 , and^#(x, false()) -> c_4() 901.15/297.03 , and^#(true(), true()) -> c_5() 901.15/297.03 , f^#(s(x)) -> c_6(h^#(x)) 901.15/297.03 , f^#(0()) -> c_7() 901.15/297.03 , h^#(s(x)) -> c_28(f^#(x)) 901.15/297.03 , h^#(0()) -> c_29() 901.15/297.03 , t^#(x) -> c_8(p^#(x, x)) 901.15/297.03 , p^#(s(x), x) -> c_19(p^#(if(gt(x, x), id(x), id(x)), s(x))) 901.15/297.03 , p^#(s(x), s(y)) -> 901.15/297.03 c_20(p^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))) 901.15/297.03 , p^#(0(), y) -> c_21(y) 901.15/297.03 , p^#(id(x), s(y)) -> c_22(p^#(x, if(gt(s(y), y), y, s(y)))) 901.15/297.03 , k^#(s(x), s(y)) -> c_9(k^#(minus(x, y), s(y))) 901.15/297.03 , k^#(0(), s(y)) -> c_10() 901.15/297.03 , minus^#(x, 0()) -> c_11(x) 901.15/297.03 , minus^#(s(x), s(y)) -> c_12(minus^#(x, y)) 901.15/297.03 , m^#(x, 0()) -> c_13(x) 901.15/297.03 , m^#(s(x), s(y)) -> c_14(m^#(x, y)) 901.15/297.03 , m^#(0(), y) -> c_15(y) 901.15/297.03 , n^#(x, 0()) -> c_16() 901.15/297.03 , n^#(s(x), s(y)) -> c_17(n^#(x, y)) 901.15/297.03 , n^#(0(), y) -> c_18() 901.15/297.03 , gt^#(s(x), s(y)) -> c_23(gt^#(x, y)) 901.15/297.03 , gt^#(s(x), 0()) -> c_24() 901.15/297.03 , gt^#(0(), y) -> c_25() 901.15/297.03 , not^#(x) -> c_26(if^#(x, false(), true())) 901.15/297.03 , id^#(x) -> c_27(x) } 901.15/297.03 Strict Trs: 901.15/297.03 { g(s(x), s(y)) -> 901.15/297.03 if(and(f(s(x)), f(s(y))), 901.15/297.03 t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), 901.15/297.03 k(n(s(x), s(y)), s(s(0()))))), 901.15/297.03 g(minus(m(x, y), n(x, y)), n(s(x), s(y)))) 901.15/297.03 , if(true(), x, y) -> x 901.15/297.03 , if(false(), x, y) -> y 901.15/297.03 , and(x, false()) -> false() 901.15/297.03 , and(true(), true()) -> true() 901.15/297.03 , f(s(x)) -> h(x) 901.15/297.03 , f(0()) -> true() 901.15/297.03 , t(x) -> p(x, x) 901.15/297.03 , k(s(x), s(y)) -> s(k(minus(x, y), s(y))) 901.15/297.03 , k(0(), s(y)) -> 0() 901.15/297.03 , minus(x, 0()) -> x 901.15/297.03 , minus(s(x), s(y)) -> minus(x, y) 901.15/297.03 , m(x, 0()) -> x 901.15/297.03 , m(s(x), s(y)) -> s(m(x, y)) 901.15/297.03 , m(0(), y) -> y 901.15/297.03 , n(x, 0()) -> 0() 901.15/297.03 , n(s(x), s(y)) -> s(n(x, y)) 901.15/297.03 , n(0(), y) -> 0() 901.15/297.03 , p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x)) 901.15/297.03 , p(s(x), s(y)) -> 901.15/297.03 s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 901.15/297.03 , p(0(), y) -> y 901.15/297.03 , p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y)))) 901.15/297.03 , gt(s(x), s(y)) -> gt(x, y) 901.15/297.03 , gt(s(x), 0()) -> true() 901.15/297.03 , gt(0(), y) -> false() 901.15/297.03 , not(x) -> if(x, false(), true()) 901.15/297.04 , id(x) -> x 901.15/297.04 , h(s(x)) -> f(x) 901.15/297.04 , h(0()) -> false() } 901.15/297.04 Obligation: 901.15/297.04 runtime complexity 901.15/297.04 Answer: 901.15/297.04 MAYBE 901.15/297.04 901.15/297.04 We estimate the number of application of {4,5,7,9,16,22,24,26,27} 901.15/297.04 by applications of Pre({4,5,7,9,16,22,24,26,27}) = 901.15/297.04 {2,3,6,8,13,15,17,19,21,23,25,29}. Here rules are labeled as 901.15/297.04 follows: 901.15/297.04 901.15/297.04 DPs: 901.15/297.04 { 1: g^#(s(x), s(y)) -> 901.15/297.04 c_1(if^#(and(f(s(x)), f(s(y))), 901.15/297.04 t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), 901.15/297.04 k(n(s(x), s(y)), s(s(0()))))), 901.15/297.04 g(minus(m(x, y), n(x, y)), n(s(x), s(y))))) 901.15/297.04 , 2: if^#(true(), x, y) -> c_2(x) 901.15/297.04 , 3: if^#(false(), x, y) -> c_3(y) 901.15/297.04 , 4: and^#(x, false()) -> c_4() 901.15/297.04 , 5: and^#(true(), true()) -> c_5() 901.15/297.04 , 6: f^#(s(x)) -> c_6(h^#(x)) 901.15/297.04 , 7: f^#(0()) -> c_7() 901.15/297.04 , 8: h^#(s(x)) -> c_28(f^#(x)) 901.15/297.04 , 9: h^#(0()) -> c_29() 901.15/297.04 , 10: t^#(x) -> c_8(p^#(x, x)) 901.15/297.04 , 11: p^#(s(x), x) -> c_19(p^#(if(gt(x, x), id(x), id(x)), s(x))) 901.15/297.04 , 12: p^#(s(x), s(y)) -> 901.15/297.04 c_20(p^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))) 901.15/297.04 , 13: p^#(0(), y) -> c_21(y) 901.15/297.04 , 14: p^#(id(x), s(y)) -> c_22(p^#(x, if(gt(s(y), y), y, s(y)))) 901.15/297.04 , 15: k^#(s(x), s(y)) -> c_9(k^#(minus(x, y), s(y))) 901.15/297.04 , 16: k^#(0(), s(y)) -> c_10() 901.15/297.04 , 17: minus^#(x, 0()) -> c_11(x) 901.15/297.04 , 18: minus^#(s(x), s(y)) -> c_12(minus^#(x, y)) 901.15/297.04 , 19: m^#(x, 0()) -> c_13(x) 901.15/297.04 , 20: m^#(s(x), s(y)) -> c_14(m^#(x, y)) 901.15/297.04 , 21: m^#(0(), y) -> c_15(y) 901.15/297.04 , 22: n^#(x, 0()) -> c_16() 901.15/297.04 , 23: n^#(s(x), s(y)) -> c_17(n^#(x, y)) 901.15/297.04 , 24: n^#(0(), y) -> c_18() 901.15/297.04 , 25: gt^#(s(x), s(y)) -> c_23(gt^#(x, y)) 901.15/297.04 , 26: gt^#(s(x), 0()) -> c_24() 901.15/297.04 , 27: gt^#(0(), y) -> c_25() 901.15/297.04 , 28: not^#(x) -> c_26(if^#(x, false(), true())) 901.15/297.04 , 29: id^#(x) -> c_27(x) } 901.15/297.04 901.15/297.04 We are left with following problem, upon which TcT provides the 901.15/297.04 certificate MAYBE. 901.15/297.04 901.15/297.04 Strict DPs: 901.15/297.04 { g^#(s(x), s(y)) -> 901.15/297.04 c_1(if^#(and(f(s(x)), f(s(y))), 901.15/297.04 t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), 901.15/297.04 k(n(s(x), s(y)), s(s(0()))))), 901.15/297.04 g(minus(m(x, y), n(x, y)), n(s(x), s(y))))) 901.15/297.04 , if^#(true(), x, y) -> c_2(x) 901.15/297.04 , if^#(false(), x, y) -> c_3(y) 901.15/297.04 , f^#(s(x)) -> c_6(h^#(x)) 901.15/297.04 , h^#(s(x)) -> c_28(f^#(x)) 901.15/297.04 , t^#(x) -> c_8(p^#(x, x)) 901.15/297.04 , p^#(s(x), x) -> c_19(p^#(if(gt(x, x), id(x), id(x)), s(x))) 901.15/297.04 , p^#(s(x), s(y)) -> 901.15/297.04 c_20(p^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))) 901.15/297.04 , p^#(0(), y) -> c_21(y) 901.15/297.04 , p^#(id(x), s(y)) -> c_22(p^#(x, if(gt(s(y), y), y, s(y)))) 901.15/297.04 , k^#(s(x), s(y)) -> c_9(k^#(minus(x, y), s(y))) 901.15/297.04 , minus^#(x, 0()) -> c_11(x) 901.15/297.04 , minus^#(s(x), s(y)) -> c_12(minus^#(x, y)) 901.15/297.04 , m^#(x, 0()) -> c_13(x) 901.15/297.04 , m^#(s(x), s(y)) -> c_14(m^#(x, y)) 901.15/297.04 , m^#(0(), y) -> c_15(y) 901.15/297.04 , n^#(s(x), s(y)) -> c_17(n^#(x, y)) 901.15/297.04 , gt^#(s(x), s(y)) -> c_23(gt^#(x, y)) 901.15/297.04 , not^#(x) -> c_26(if^#(x, false(), true())) 901.15/297.04 , id^#(x) -> c_27(x) } 901.15/297.04 Strict Trs: 901.15/297.04 { g(s(x), s(y)) -> 901.15/297.04 if(and(f(s(x)), f(s(y))), 901.15/297.04 t(g(k(minus(m(x, y), n(x, y)), s(s(0()))), 901.15/297.04 k(n(s(x), s(y)), s(s(0()))))), 901.15/297.04 g(minus(m(x, y), n(x, y)), n(s(x), s(y)))) 901.15/297.04 , if(true(), x, y) -> x 901.15/297.04 , if(false(), x, y) -> y 901.15/297.04 , and(x, false()) -> false() 901.15/297.04 , and(true(), true()) -> true() 901.15/297.04 , f(s(x)) -> h(x) 901.15/297.04 , f(0()) -> true() 901.15/297.04 , t(x) -> p(x, x) 901.15/297.04 , k(s(x), s(y)) -> s(k(minus(x, y), s(y))) 901.15/297.04 , k(0(), s(y)) -> 0() 901.15/297.04 , minus(x, 0()) -> x 901.15/297.04 , minus(s(x), s(y)) -> minus(x, y) 901.15/297.04 , m(x, 0()) -> x 901.15/297.04 , m(s(x), s(y)) -> s(m(x, y)) 901.15/297.04 , m(0(), y) -> y 901.15/297.04 , n(x, 0()) -> 0() 901.15/297.04 , n(s(x), s(y)) -> s(n(x, y)) 901.15/297.04 , n(0(), y) -> 0() 901.15/297.04 , p(s(x), x) -> p(if(gt(x, x), id(x), id(x)), s(x)) 901.15/297.04 , p(s(x), s(y)) -> 901.15/297.04 s(s(p(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) 901.15/297.04 , p(0(), y) -> y 901.15/297.04 , p(id(x), s(y)) -> s(p(x, if(gt(s(y), y), y, s(y)))) 901.15/297.04 , gt(s(x), s(y)) -> gt(x, y) 901.15/297.04 , gt(s(x), 0()) -> true() 901.15/297.04 , gt(0(), y) -> false() 901.15/297.04 , not(x) -> if(x, false(), true()) 901.15/297.04 , id(x) -> x 901.15/297.04 , h(s(x)) -> f(x) 901.15/297.04 , h(0()) -> false() } 901.15/297.04 Weak DPs: 901.15/297.04 { and^#(x, false()) -> c_4() 901.15/297.04 , and^#(true(), true()) -> c_5() 901.15/297.04 , f^#(0()) -> c_7() 901.15/297.04 , h^#(0()) -> c_29() 901.15/297.04 , k^#(0(), s(y)) -> c_10() 901.15/297.04 , n^#(x, 0()) -> c_16() 901.15/297.04 , n^#(0(), y) -> c_18() 901.15/297.04 , gt^#(s(x), 0()) -> c_24() 901.15/297.04 , gt^#(0(), y) -> c_25() } 901.15/297.04 Obligation: 901.15/297.04 runtime complexity 901.15/297.04 Answer: 901.15/297.04 MAYBE 901.15/297.04 901.15/297.04 Empty strict component of the problem is NOT empty. 901.15/297.04 901.15/297.04 901.15/297.04 Arrrr.. 901.28/297.17 EOF