MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , zero(0()) -> true() , zero(s(x)) -> false() , id(0()) -> 0() , id(s(x)) -> s(id(x)) , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) , mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) , if_mod(true(), b1, b2, x, y) -> 0() , if_mod(false(), b1, b2, x, y) -> if2(b1, b2, x, y) , if2(true(), b2, x, y) -> 0() , if2(false(), b2, x, y) -> if3(b2, x, y) , if3(true(), x, y) -> mod(minus(x, y), s(y)) , if3(false(), x, y) -> 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: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Fastest' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 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 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) '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: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , le^#(s(x), s(y)) -> c_3(le^#(x, y)) , zero^#(0()) -> c_4() , zero^#(s(x)) -> c_5() , id^#(0()) -> c_6() , id^#(s(x)) -> c_7(id^#(x)) , minus^#(x, 0()) -> c_8(x) , minus^#(s(x), s(y)) -> c_9(minus^#(x, y)) , mod^#(x, y) -> c_10(if_mod^#(zero(x), zero(y), le(y, x), id(x), id(y))) , if_mod^#(true(), b1, b2, x, y) -> c_11() , if_mod^#(false(), b1, b2, x, y) -> c_12(if2^#(b1, b2, x, y)) , if2^#(true(), b2, x, y) -> c_13() , if2^#(false(), b2, x, y) -> c_14(if3^#(b2, x, y)) , if3^#(true(), x, y) -> c_15(mod^#(minus(x, y), s(y))) , if3^#(false(), x, y) -> c_16(x) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , le^#(s(x), s(y)) -> c_3(le^#(x, y)) , zero^#(0()) -> c_4() , zero^#(s(x)) -> c_5() , id^#(0()) -> c_6() , id^#(s(x)) -> c_7(id^#(x)) , minus^#(x, 0()) -> c_8(x) , minus^#(s(x), s(y)) -> c_9(minus^#(x, y)) , mod^#(x, y) -> c_10(if_mod^#(zero(x), zero(y), le(y, x), id(x), id(y))) , if_mod^#(true(), b1, b2, x, y) -> c_11() , if_mod^#(false(), b1, b2, x, y) -> c_12(if2^#(b1, b2, x, y)) , if2^#(true(), b2, x, y) -> c_13() , if2^#(false(), b2, x, y) -> c_14(if3^#(b2, x, y)) , if3^#(true(), x, y) -> c_15(mod^#(minus(x, y), s(y))) , if3^#(false(), x, y) -> c_16(x) } Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , zero(0()) -> true() , zero(s(x)) -> false() , id(0()) -> 0() , id(s(x)) -> s(id(x)) , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) , mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) , if_mod(true(), b1, b2, x, y) -> 0() , if_mod(false(), b1, b2, x, y) -> if2(b1, b2, x, y) , if2(true(), b2, x, y) -> 0() , if2(false(), b2, x, y) -> if3(b2, x, y) , if3(true(), x, y) -> mod(minus(x, y), s(y)) , if3(false(), x, y) -> x } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,4,5,6,11,13} by applications of Pre({1,2,4,5,6,11,13}) = {3,7,8,10,12,16}. Here rules are labeled as follows: DPs: { 1: le^#(0(), y) -> c_1() , 2: le^#(s(x), 0()) -> c_2() , 3: le^#(s(x), s(y)) -> c_3(le^#(x, y)) , 4: zero^#(0()) -> c_4() , 5: zero^#(s(x)) -> c_5() , 6: id^#(0()) -> c_6() , 7: id^#(s(x)) -> c_7(id^#(x)) , 8: minus^#(x, 0()) -> c_8(x) , 9: minus^#(s(x), s(y)) -> c_9(minus^#(x, y)) , 10: mod^#(x, y) -> c_10(if_mod^#(zero(x), zero(y), le(y, x), id(x), id(y))) , 11: if_mod^#(true(), b1, b2, x, y) -> c_11() , 12: if_mod^#(false(), b1, b2, x, y) -> c_12(if2^#(b1, b2, x, y)) , 13: if2^#(true(), b2, x, y) -> c_13() , 14: if2^#(false(), b2, x, y) -> c_14(if3^#(b2, x, y)) , 15: if3^#(true(), x, y) -> c_15(mod^#(minus(x, y), s(y))) , 16: if3^#(false(), x, y) -> c_16(x) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(s(x), s(y)) -> c_3(le^#(x, y)) , id^#(s(x)) -> c_7(id^#(x)) , minus^#(x, 0()) -> c_8(x) , minus^#(s(x), s(y)) -> c_9(minus^#(x, y)) , mod^#(x, y) -> c_10(if_mod^#(zero(x), zero(y), le(y, x), id(x), id(y))) , if_mod^#(false(), b1, b2, x, y) -> c_12(if2^#(b1, b2, x, y)) , if2^#(false(), b2, x, y) -> c_14(if3^#(b2, x, y)) , if3^#(true(), x, y) -> c_15(mod^#(minus(x, y), s(y))) , if3^#(false(), x, y) -> c_16(x) } Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , zero(0()) -> true() , zero(s(x)) -> false() , id(0()) -> 0() , id(s(x)) -> s(id(x)) , minus(x, 0()) -> x , minus(s(x), s(y)) -> minus(x, y) , mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y)) , if_mod(true(), b1, b2, x, y) -> 0() , if_mod(false(), b1, b2, x, y) -> if2(b1, b2, x, y) , if2(true(), b2, x, y) -> 0() , if2(false(), b2, x, y) -> if3(b2, x, y) , if3(true(), x, y) -> mod(minus(x, y), s(y)) , if3(false(), x, y) -> x } Weak DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , zero^#(0()) -> c_4() , zero^#(s(x)) -> c_5() , id^#(0()) -> c_6() , if_mod^#(true(), b1, b2, x, y) -> c_11() , if2^#(true(), b2, x, y) -> c_13() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..