MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z))) , times(x, s(y)) -> plus(times(x, y), x) , times(x, 0()) -> 0() , plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) , plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) , plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) , plus(zero(), y) -> y , if(true(), x, y) -> x , if(false(), x, y) -> y , gt(s(x), s(y)) -> gt(x, y) , gt(s(x), zero()) -> true() , gt(zero(), y) -> false() , not(x) -> if(x, false(), true()) , id(x) -> 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: Computation stopped due to timeout after 30.0 seconds. 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 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) '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: { times^#(x, plus(y, s(z))) -> c_1(plus^#(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))) , times^#(x, s(y)) -> c_2(plus^#(times(x, y), x)) , times^#(x, 0()) -> c_3() , plus^#(s(x), x) -> c_4(plus^#(if(gt(x, x), id(x), id(x)), s(x))) , plus^#(s(x), s(y)) -> c_5(plus^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))) , plus^#(id(x), s(y)) -> c_6(plus^#(x, if(gt(s(y), y), y, s(y)))) , plus^#(zero(), y) -> c_7(y) , if^#(true(), x, y) -> c_8(x) , if^#(false(), x, y) -> c_9(y) , gt^#(s(x), s(y)) -> c_10(gt^#(x, y)) , gt^#(s(x), zero()) -> c_11() , gt^#(zero(), y) -> c_12() , not^#(x) -> c_13(if^#(x, false(), true())) , id^#(x) -> c_14(x) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { times^#(x, plus(y, s(z))) -> c_1(plus^#(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))) , times^#(x, s(y)) -> c_2(plus^#(times(x, y), x)) , times^#(x, 0()) -> c_3() , plus^#(s(x), x) -> c_4(plus^#(if(gt(x, x), id(x), id(x)), s(x))) , plus^#(s(x), s(y)) -> c_5(plus^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))) , plus^#(id(x), s(y)) -> c_6(plus^#(x, if(gt(s(y), y), y, s(y)))) , plus^#(zero(), y) -> c_7(y) , if^#(true(), x, y) -> c_8(x) , if^#(false(), x, y) -> c_9(y) , gt^#(s(x), s(y)) -> c_10(gt^#(x, y)) , gt^#(s(x), zero()) -> c_11() , gt^#(zero(), y) -> c_12() , not^#(x) -> c_13(if^#(x, false(), true())) , id^#(x) -> c_14(x) } Strict Trs: { times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z))) , times(x, s(y)) -> plus(times(x, y), x) , times(x, 0()) -> 0() , plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) , plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) , plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) , plus(zero(), y) -> y , if(true(), x, y) -> x , if(false(), x, y) -> y , gt(s(x), s(y)) -> gt(x, y) , gt(s(x), zero()) -> true() , gt(zero(), y) -> false() , not(x) -> if(x, false(), true()) , id(x) -> x } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,11,12} by applications of Pre({3,11,12}) = {7,8,9,10,14}. Here rules are labeled as follows: DPs: { 1: times^#(x, plus(y, s(z))) -> c_1(plus^#(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))) , 2: times^#(x, s(y)) -> c_2(plus^#(times(x, y), x)) , 3: times^#(x, 0()) -> c_3() , 4: plus^#(s(x), x) -> c_4(plus^#(if(gt(x, x), id(x), id(x)), s(x))) , 5: plus^#(s(x), s(y)) -> c_5(plus^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))) , 6: plus^#(id(x), s(y)) -> c_6(plus^#(x, if(gt(s(y), y), y, s(y)))) , 7: plus^#(zero(), y) -> c_7(y) , 8: if^#(true(), x, y) -> c_8(x) , 9: if^#(false(), x, y) -> c_9(y) , 10: gt^#(s(x), s(y)) -> c_10(gt^#(x, y)) , 11: gt^#(s(x), zero()) -> c_11() , 12: gt^#(zero(), y) -> c_12() , 13: not^#(x) -> c_13(if^#(x, false(), true())) , 14: id^#(x) -> c_14(x) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { times^#(x, plus(y, s(z))) -> c_1(plus^#(times(x, plus(y, times(s(z), 0()))), times(x, s(z)))) , times^#(x, s(y)) -> c_2(plus^#(times(x, y), x)) , plus^#(s(x), x) -> c_4(plus^#(if(gt(x, x), id(x), id(x)), s(x))) , plus^#(s(x), s(y)) -> c_5(plus^#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))) , plus^#(id(x), s(y)) -> c_6(plus^#(x, if(gt(s(y), y), y, s(y)))) , plus^#(zero(), y) -> c_7(y) , if^#(true(), x, y) -> c_8(x) , if^#(false(), x, y) -> c_9(y) , gt^#(s(x), s(y)) -> c_10(gt^#(x, y)) , not^#(x) -> c_13(if^#(x, false(), true())) , id^#(x) -> c_14(x) } Strict Trs: { times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0()))), times(x, s(z))) , times(x, s(y)) -> plus(times(x, y), x) , times(x, 0()) -> 0() , plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)) , plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) , plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))) , plus(zero(), y) -> y , if(true(), x, y) -> x , if(false(), x, y) -> y , gt(s(x), s(y)) -> gt(x, y) , gt(s(x), zero()) -> true() , gt(zero(), y) -> false() , not(x) -> if(x, false(), true()) , id(x) -> x } Weak DPs: { times^#(x, 0()) -> c_3() , gt^#(s(x), zero()) -> c_11() , gt^#(zero(), y) -> c_12() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..