MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { p(0()) -> 0() , p(s(x)) -> x , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, p(s(y)))) , plus(0(), y) -> y , plus(s(x), y) -> s(plus(x, y)) , plus(s(x), y) -> s(plus(p(s(x)), y)) , times(0(), y) -> 0() , times(s(x), y) -> plus(y, times(x, y)) , times(s(0()), y) -> y , div(x, y) -> quot(x, y, y) , div(0(), y) -> 0() , div(div(x, y), z) -> div(x, times(y, z)) , quot(x, 0(), s(z)) -> s(div(x, s(z))) , quot(0(), s(y), z) -> 0() , quot(s(x), s(y), z) -> quot(x, y, z) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , divides(y, x) -> eq(x, times(div(x, y), y)) , prime(s(s(x))) -> pr(s(s(x)), s(x)) , pr(x, s(0())) -> true() , pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) , if(true(), x, y) -> false() , if(false(), x, y) -> pr(x, y) } 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 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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { p^#(0()) -> c_1() , p^#(s(x)) -> c_2(x) , plus^#(x, 0()) -> c_3(x) , plus^#(x, s(y)) -> c_4(plus^#(x, p(s(y)))) , plus^#(0(), y) -> c_5(y) , plus^#(s(x), y) -> c_6(plus^#(x, y)) , plus^#(s(x), y) -> c_7(plus^#(p(s(x)), y)) , times^#(0(), y) -> c_8() , times^#(s(x), y) -> c_9(plus^#(y, times(x, y))) , times^#(s(0()), y) -> c_10(y) , div^#(x, y) -> c_11(quot^#(x, y, y)) , div^#(0(), y) -> c_12() , div^#(div(x, y), z) -> c_13(div^#(x, times(y, z))) , quot^#(x, 0(), s(z)) -> c_14(div^#(x, s(z))) , quot^#(0(), s(y), z) -> c_15() , quot^#(s(x), s(y), z) -> c_16(quot^#(x, y, z)) , eq^#(0(), 0()) -> c_17() , eq^#(0(), s(y)) -> c_18() , eq^#(s(x), 0()) -> c_19() , eq^#(s(x), s(y)) -> c_20(eq^#(x, y)) , divides^#(y, x) -> c_21(eq^#(x, times(div(x, y), y))) , prime^#(s(s(x))) -> c_22(pr^#(s(s(x)), s(x))) , pr^#(x, s(0())) -> c_23() , pr^#(x, s(s(y))) -> c_24(if^#(divides(s(s(y)), x), x, s(y))) , if^#(true(), x, y) -> c_25() , if^#(false(), x, y) -> c_26(pr^#(x, y)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { p^#(0()) -> c_1() , p^#(s(x)) -> c_2(x) , plus^#(x, 0()) -> c_3(x) , plus^#(x, s(y)) -> c_4(plus^#(x, p(s(y)))) , plus^#(0(), y) -> c_5(y) , plus^#(s(x), y) -> c_6(plus^#(x, y)) , plus^#(s(x), y) -> c_7(plus^#(p(s(x)), y)) , times^#(0(), y) -> c_8() , times^#(s(x), y) -> c_9(plus^#(y, times(x, y))) , times^#(s(0()), y) -> c_10(y) , div^#(x, y) -> c_11(quot^#(x, y, y)) , div^#(0(), y) -> c_12() , div^#(div(x, y), z) -> c_13(div^#(x, times(y, z))) , quot^#(x, 0(), s(z)) -> c_14(div^#(x, s(z))) , quot^#(0(), s(y), z) -> c_15() , quot^#(s(x), s(y), z) -> c_16(quot^#(x, y, z)) , eq^#(0(), 0()) -> c_17() , eq^#(0(), s(y)) -> c_18() , eq^#(s(x), 0()) -> c_19() , eq^#(s(x), s(y)) -> c_20(eq^#(x, y)) , divides^#(y, x) -> c_21(eq^#(x, times(div(x, y), y))) , prime^#(s(s(x))) -> c_22(pr^#(s(s(x)), s(x))) , pr^#(x, s(0())) -> c_23() , pr^#(x, s(s(y))) -> c_24(if^#(divides(s(s(y)), x), x, s(y))) , if^#(true(), x, y) -> c_25() , if^#(false(), x, y) -> c_26(pr^#(x, y)) } Strict Trs: { p(0()) -> 0() , p(s(x)) -> x , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, p(s(y)))) , plus(0(), y) -> y , plus(s(x), y) -> s(plus(x, y)) , plus(s(x), y) -> s(plus(p(s(x)), y)) , times(0(), y) -> 0() , times(s(x), y) -> plus(y, times(x, y)) , times(s(0()), y) -> y , div(x, y) -> quot(x, y, y) , div(0(), y) -> 0() , div(div(x, y), z) -> div(x, times(y, z)) , quot(x, 0(), s(z)) -> s(div(x, s(z))) , quot(0(), s(y), z) -> 0() , quot(s(x), s(y), z) -> quot(x, y, z) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , divides(y, x) -> eq(x, times(div(x, y), y)) , prime(s(s(x))) -> pr(s(s(x)), s(x)) , pr(x, s(0())) -> true() , pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) , if(true(), x, y) -> false() , if(false(), x, y) -> pr(x, y) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,8,12,15,17,18,19,23,25} by applications of Pre({1,8,12,15,17,18,19,23,25}) = {2,3,5,10,11,13,14,16,20,21,22,24,26}. Here rules are labeled as follows: DPs: { 1: p^#(0()) -> c_1() , 2: p^#(s(x)) -> c_2(x) , 3: plus^#(x, 0()) -> c_3(x) , 4: plus^#(x, s(y)) -> c_4(plus^#(x, p(s(y)))) , 5: plus^#(0(), y) -> c_5(y) , 6: plus^#(s(x), y) -> c_6(plus^#(x, y)) , 7: plus^#(s(x), y) -> c_7(plus^#(p(s(x)), y)) , 8: times^#(0(), y) -> c_8() , 9: times^#(s(x), y) -> c_9(plus^#(y, times(x, y))) , 10: times^#(s(0()), y) -> c_10(y) , 11: div^#(x, y) -> c_11(quot^#(x, y, y)) , 12: div^#(0(), y) -> c_12() , 13: div^#(div(x, y), z) -> c_13(div^#(x, times(y, z))) , 14: quot^#(x, 0(), s(z)) -> c_14(div^#(x, s(z))) , 15: quot^#(0(), s(y), z) -> c_15() , 16: quot^#(s(x), s(y), z) -> c_16(quot^#(x, y, z)) , 17: eq^#(0(), 0()) -> c_17() , 18: eq^#(0(), s(y)) -> c_18() , 19: eq^#(s(x), 0()) -> c_19() , 20: eq^#(s(x), s(y)) -> c_20(eq^#(x, y)) , 21: divides^#(y, x) -> c_21(eq^#(x, times(div(x, y), y))) , 22: prime^#(s(s(x))) -> c_22(pr^#(s(s(x)), s(x))) , 23: pr^#(x, s(0())) -> c_23() , 24: pr^#(x, s(s(y))) -> c_24(if^#(divides(s(s(y)), x), x, s(y))) , 25: if^#(true(), x, y) -> c_25() , 26: if^#(false(), x, y) -> c_26(pr^#(x, y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { p^#(s(x)) -> c_2(x) , plus^#(x, 0()) -> c_3(x) , plus^#(x, s(y)) -> c_4(plus^#(x, p(s(y)))) , plus^#(0(), y) -> c_5(y) , plus^#(s(x), y) -> c_6(plus^#(x, y)) , plus^#(s(x), y) -> c_7(plus^#(p(s(x)), y)) , times^#(s(x), y) -> c_9(plus^#(y, times(x, y))) , times^#(s(0()), y) -> c_10(y) , div^#(x, y) -> c_11(quot^#(x, y, y)) , div^#(div(x, y), z) -> c_13(div^#(x, times(y, z))) , quot^#(x, 0(), s(z)) -> c_14(div^#(x, s(z))) , quot^#(s(x), s(y), z) -> c_16(quot^#(x, y, z)) , eq^#(s(x), s(y)) -> c_20(eq^#(x, y)) , divides^#(y, x) -> c_21(eq^#(x, times(div(x, y), y))) , prime^#(s(s(x))) -> c_22(pr^#(s(s(x)), s(x))) , pr^#(x, s(s(y))) -> c_24(if^#(divides(s(s(y)), x), x, s(y))) , if^#(false(), x, y) -> c_26(pr^#(x, y)) } Strict Trs: { p(0()) -> 0() , p(s(x)) -> x , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, p(s(y)))) , plus(0(), y) -> y , plus(s(x), y) -> s(plus(x, y)) , plus(s(x), y) -> s(plus(p(s(x)), y)) , times(0(), y) -> 0() , times(s(x), y) -> plus(y, times(x, y)) , times(s(0()), y) -> y , div(x, y) -> quot(x, y, y) , div(0(), y) -> 0() , div(div(x, y), z) -> div(x, times(y, z)) , quot(x, 0(), s(z)) -> s(div(x, s(z))) , quot(0(), s(y), z) -> 0() , quot(s(x), s(y), z) -> quot(x, y, z) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , divides(y, x) -> eq(x, times(div(x, y), y)) , prime(s(s(x))) -> pr(s(s(x)), s(x)) , pr(x, s(0())) -> true() , pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y)) , if(true(), x, y) -> false() , if(false(), x, y) -> pr(x, y) } Weak DPs: { p^#(0()) -> c_1() , times^#(0(), y) -> c_8() , div^#(0(), y) -> c_12() , quot^#(0(), s(y), z) -> c_15() , eq^#(0(), 0()) -> c_17() , eq^#(0(), s(y)) -> c_18() , eq^#(s(x), 0()) -> c_19() , pr^#(x, s(0())) -> c_23() , if^#(true(), x, y) -> c_25() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..