MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { minus(s(x), y) -> if(gt(s(x), y), x, y) , if(true(), x, y) -> s(minus(x, y)) , if(false(), x, y) -> 0() , gt(s(x), s(y)) -> gt(x, y) , gt(s(x), 0()) -> true() , gt(0(), y) -> false() , ge(x, 0()) -> true() , ge(s(x), s(y)) -> ge(x, y) , ge(0(), s(x)) -> false() , div(x, y) -> if1(ge(x, y), x, y) , if1(true(), x, y) -> if2(gt(y, 0()), x, y) , if1(false(), x, y) -> 0() , if2(true(), x, y) -> s(div(minus(x, y), y)) , if2(false(), x, y) -> 0() } 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 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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { minus^#(s(x), y) -> c_1(if^#(gt(s(x), y), x, y)) , if^#(true(), x, y) -> c_2(minus^#(x, y)) , if^#(false(), x, y) -> c_3() , gt^#(s(x), s(y)) -> c_4(gt^#(x, y)) , gt^#(s(x), 0()) -> c_5() , gt^#(0(), y) -> c_6() , ge^#(x, 0()) -> c_7() , ge^#(s(x), s(y)) -> c_8(ge^#(x, y)) , ge^#(0(), s(x)) -> c_9() , div^#(x, y) -> c_10(if1^#(ge(x, y), x, y)) , if1^#(true(), x, y) -> c_11(if2^#(gt(y, 0()), x, y)) , if1^#(false(), x, y) -> c_12() , if2^#(true(), x, y) -> c_13(div^#(minus(x, y), y)) , if2^#(false(), x, y) -> c_14() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(s(x), y) -> c_1(if^#(gt(s(x), y), x, y)) , if^#(true(), x, y) -> c_2(minus^#(x, y)) , if^#(false(), x, y) -> c_3() , gt^#(s(x), s(y)) -> c_4(gt^#(x, y)) , gt^#(s(x), 0()) -> c_5() , gt^#(0(), y) -> c_6() , ge^#(x, 0()) -> c_7() , ge^#(s(x), s(y)) -> c_8(ge^#(x, y)) , ge^#(0(), s(x)) -> c_9() , div^#(x, y) -> c_10(if1^#(ge(x, y), x, y)) , if1^#(true(), x, y) -> c_11(if2^#(gt(y, 0()), x, y)) , if1^#(false(), x, y) -> c_12() , if2^#(true(), x, y) -> c_13(div^#(minus(x, y), y)) , if2^#(false(), x, y) -> c_14() } Strict Trs: { minus(s(x), y) -> if(gt(s(x), y), x, y) , if(true(), x, y) -> s(minus(x, y)) , if(false(), x, y) -> 0() , gt(s(x), s(y)) -> gt(x, y) , gt(s(x), 0()) -> true() , gt(0(), y) -> false() , ge(x, 0()) -> true() , ge(s(x), s(y)) -> ge(x, y) , ge(0(), s(x)) -> false() , div(x, y) -> if1(ge(x, y), x, y) , if1(true(), x, y) -> if2(gt(y, 0()), x, y) , if1(false(), x, y) -> 0() , if2(true(), x, y) -> s(div(minus(x, y), y)) , if2(false(), x, y) -> 0() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,5,6,7,9,12,14} by applications of Pre({3,5,6,7,9,12,14}) = {1,4,8,10,11}. Here rules are labeled as follows: DPs: { 1: minus^#(s(x), y) -> c_1(if^#(gt(s(x), y), x, y)) , 2: if^#(true(), x, y) -> c_2(minus^#(x, y)) , 3: if^#(false(), x, y) -> c_3() , 4: gt^#(s(x), s(y)) -> c_4(gt^#(x, y)) , 5: gt^#(s(x), 0()) -> c_5() , 6: gt^#(0(), y) -> c_6() , 7: ge^#(x, 0()) -> c_7() , 8: ge^#(s(x), s(y)) -> c_8(ge^#(x, y)) , 9: ge^#(0(), s(x)) -> c_9() , 10: div^#(x, y) -> c_10(if1^#(ge(x, y), x, y)) , 11: if1^#(true(), x, y) -> c_11(if2^#(gt(y, 0()), x, y)) , 12: if1^#(false(), x, y) -> c_12() , 13: if2^#(true(), x, y) -> c_13(div^#(minus(x, y), y)) , 14: if2^#(false(), x, y) -> c_14() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(s(x), y) -> c_1(if^#(gt(s(x), y), x, y)) , if^#(true(), x, y) -> c_2(minus^#(x, y)) , gt^#(s(x), s(y)) -> c_4(gt^#(x, y)) , ge^#(s(x), s(y)) -> c_8(ge^#(x, y)) , div^#(x, y) -> c_10(if1^#(ge(x, y), x, y)) , if1^#(true(), x, y) -> c_11(if2^#(gt(y, 0()), x, y)) , if2^#(true(), x, y) -> c_13(div^#(minus(x, y), y)) } Strict Trs: { minus(s(x), y) -> if(gt(s(x), y), x, y) , if(true(), x, y) -> s(minus(x, y)) , if(false(), x, y) -> 0() , gt(s(x), s(y)) -> gt(x, y) , gt(s(x), 0()) -> true() , gt(0(), y) -> false() , ge(x, 0()) -> true() , ge(s(x), s(y)) -> ge(x, y) , ge(0(), s(x)) -> false() , div(x, y) -> if1(ge(x, y), x, y) , if1(true(), x, y) -> if2(gt(y, 0()), x, y) , if1(false(), x, y) -> 0() , if2(true(), x, y) -> s(div(minus(x, y), y)) , if2(false(), x, y) -> 0() } Weak DPs: { if^#(false(), x, y) -> c_3() , gt^#(s(x), 0()) -> c_5() , gt^#(0(), y) -> c_6() , ge^#(x, 0()) -> c_7() , ge^#(0(), s(x)) -> c_9() , if1^#(false(), x, y) -> c_12() , if2^#(false(), x, y) -> c_14() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..