MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { minus(0(), y) -> 0() , 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(0(), y) -> false() , gt(s(x), 0()) -> true() , gt(s(x), s(y)) -> gt(x, y) , mod(x, 0()) -> 0() , mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y)) , if1(true(), x, y) -> x , if1(false(), x, y) -> mod(minus(x, y), y) , lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) '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. 2) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { minus^#(0(), y) -> c_1() , minus^#(s(x), y) -> c_2(if^#(gt(s(x), y), x, y)) , if^#(true(), x, y) -> c_3(minus^#(x, y)) , if^#(false(), x, y) -> c_4() , gt^#(0(), y) -> c_5() , gt^#(s(x), 0()) -> c_6() , gt^#(s(x), s(y)) -> c_7(gt^#(x, y)) , mod^#(x, 0()) -> c_8() , mod^#(x, s(y)) -> c_9(if1^#(lt(x, s(y)), x, s(y))) , if1^#(true(), x, y) -> c_10(x) , if1^#(false(), x, y) -> c_11(mod^#(minus(x, y), y)) , lt^#(x, 0()) -> c_12() , lt^#(0(), s(x)) -> c_13() , lt^#(s(x), s(y)) -> c_14(lt^#(x, y)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(0(), y) -> c_1() , minus^#(s(x), y) -> c_2(if^#(gt(s(x), y), x, y)) , if^#(true(), x, y) -> c_3(minus^#(x, y)) , if^#(false(), x, y) -> c_4() , gt^#(0(), y) -> c_5() , gt^#(s(x), 0()) -> c_6() , gt^#(s(x), s(y)) -> c_7(gt^#(x, y)) , mod^#(x, 0()) -> c_8() , mod^#(x, s(y)) -> c_9(if1^#(lt(x, s(y)), x, s(y))) , if1^#(true(), x, y) -> c_10(x) , if1^#(false(), x, y) -> c_11(mod^#(minus(x, y), y)) , lt^#(x, 0()) -> c_12() , lt^#(0(), s(x)) -> c_13() , lt^#(s(x), s(y)) -> c_14(lt^#(x, y)) } Strict Trs: { minus(0(), y) -> 0() , 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(0(), y) -> false() , gt(s(x), 0()) -> true() , gt(s(x), s(y)) -> gt(x, y) , mod(x, 0()) -> 0() , mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y)) , if1(true(), x, y) -> x , if1(false(), x, y) -> mod(minus(x, y), y) , lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,4,5,6,8,12,13} by applications of Pre({1,4,5,6,8,12,13}) = {2,3,7,10,11,14}. Here rules are labeled as follows: DPs: { 1: minus^#(0(), y) -> c_1() , 2: minus^#(s(x), y) -> c_2(if^#(gt(s(x), y), x, y)) , 3: if^#(true(), x, y) -> c_3(minus^#(x, y)) , 4: if^#(false(), x, y) -> c_4() , 5: gt^#(0(), y) -> c_5() , 6: gt^#(s(x), 0()) -> c_6() , 7: gt^#(s(x), s(y)) -> c_7(gt^#(x, y)) , 8: mod^#(x, 0()) -> c_8() , 9: mod^#(x, s(y)) -> c_9(if1^#(lt(x, s(y)), x, s(y))) , 10: if1^#(true(), x, y) -> c_10(x) , 11: if1^#(false(), x, y) -> c_11(mod^#(minus(x, y), y)) , 12: lt^#(x, 0()) -> c_12() , 13: lt^#(0(), s(x)) -> c_13() , 14: lt^#(s(x), s(y)) -> c_14(lt^#(x, y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(s(x), y) -> c_2(if^#(gt(s(x), y), x, y)) , if^#(true(), x, y) -> c_3(minus^#(x, y)) , gt^#(s(x), s(y)) -> c_7(gt^#(x, y)) , mod^#(x, s(y)) -> c_9(if1^#(lt(x, s(y)), x, s(y))) , if1^#(true(), x, y) -> c_10(x) , if1^#(false(), x, y) -> c_11(mod^#(minus(x, y), y)) , lt^#(s(x), s(y)) -> c_14(lt^#(x, y)) } Strict Trs: { minus(0(), y) -> 0() , 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(0(), y) -> false() , gt(s(x), 0()) -> true() , gt(s(x), s(y)) -> gt(x, y) , mod(x, 0()) -> 0() , mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y)) , if1(true(), x, y) -> x , if1(false(), x, y) -> mod(minus(x, y), y) , lt(x, 0()) -> false() , lt(0(), s(x)) -> true() , lt(s(x), s(y)) -> lt(x, y) } Weak DPs: { minus^#(0(), y) -> c_1() , if^#(false(), x, y) -> c_4() , gt^#(0(), y) -> c_5() , gt^#(s(x), 0()) -> c_6() , mod^#(x, 0()) -> c_8() , lt^#(x, 0()) -> c_12() , lt^#(0(), s(x)) -> c_13() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..