MAYBE 1188.36/298.10 MAYBE 1188.36/298.10 1188.36/298.10 We are left with following problem, upon which TcT provides the 1188.36/298.10 certificate MAYBE. 1188.36/298.10 1188.36/298.10 Strict Trs: 1188.36/298.10 { a__primes() -> a__sieve(a__from(s(s(0())))) 1188.36/298.10 , a__primes() -> primes() 1188.36/298.10 , a__sieve(X) -> sieve(X) 1188.36/298.10 , a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) 1188.36/298.10 , a__from(X) -> cons(mark(X), from(s(X))) 1188.36/298.10 , a__from(X) -> from(X) 1188.36/298.10 , mark(s(X)) -> s(mark(X)) 1188.36/298.10 , mark(0()) -> 0() 1188.36/298.10 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 1188.36/298.10 , mark(from(X)) -> a__from(mark(X)) 1188.36/298.10 , mark(true()) -> true() 1188.36/298.10 , mark(false()) -> false() 1188.36/298.10 , mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) 1188.36/298.10 , mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) 1188.36/298.10 , mark(sieve(X)) -> a__sieve(mark(X)) 1188.36/298.10 , mark(primes()) -> a__primes() 1188.36/298.10 , mark(head(X)) -> a__head(mark(X)) 1188.36/298.10 , mark(tail(X)) -> a__tail(mark(X)) 1188.36/298.10 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1188.36/298.10 , a__head(X) -> head(X) 1188.36/298.10 , a__head(cons(X, Y)) -> mark(X) 1188.36/298.10 , a__tail(X) -> tail(X) 1188.36/298.10 , a__tail(cons(X, Y)) -> mark(Y) 1188.36/298.10 , a__if(X1, X2, X3) -> if(X1, X2, X3) 1188.36/298.10 , a__if(true(), X, Y) -> mark(X) 1188.36/298.10 , a__if(false(), X, Y) -> mark(Y) 1188.36/298.10 , a__filter(X1, X2) -> filter(X1, X2) 1188.36/298.10 , a__filter(s(s(X)), cons(Y, Z)) -> 1188.36/298.10 a__if(divides(s(s(mark(X))), mark(Y)), 1188.36/298.10 filter(s(s(X)), Z), 1188.36/298.10 cons(Y, filter(X, sieve(Y)))) } 1188.36/298.10 Obligation: 1188.36/298.10 runtime complexity 1188.36/298.10 Answer: 1188.36/298.10 MAYBE 1188.36/298.10 1188.36/298.10 None of the processors succeeded. 1188.36/298.10 1188.36/298.10 Details of failed attempt(s): 1188.36/298.10 ----------------------------- 1188.36/298.10 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 1188.36/298.10 following reason: 1188.36/298.10 1188.36/298.10 Computation stopped due to timeout after 297.0 seconds. 1188.36/298.10 1188.36/298.10 2) 'Best' failed due to the following reason: 1188.36/298.10 1188.36/298.10 None of the processors succeeded. 1188.36/298.10 1188.36/298.10 Details of failed attempt(s): 1188.36/298.10 ----------------------------- 1188.36/298.10 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 1188.36/298.10 seconds)' failed due to the following reason: 1188.36/298.10 1188.36/298.10 Computation stopped due to timeout after 148.0 seconds. 1188.36/298.10 1188.36/298.10 2) 'Best' failed due to the following reason: 1188.36/298.10 1188.36/298.10 None of the processors succeeded. 1188.36/298.10 1188.36/298.10 Details of failed attempt(s): 1188.36/298.11 ----------------------------- 1188.36/298.11 1) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 1188.36/298.11 to the following reason: 1188.36/298.11 1188.36/298.11 The processor is inapplicable, reason: 1188.36/298.11 Processor only applicable for innermost runtime complexity analysis 1188.36/298.11 1188.36/298.11 2) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 1188.36/298.11 following reason: 1188.36/298.11 1188.36/298.11 The processor is inapplicable, reason: 1188.36/298.11 Processor only applicable for innermost runtime complexity analysis 1188.36/298.11 1188.36/298.11 1188.36/298.11 3) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 1188.36/298.11 failed due to the following reason: 1188.36/298.11 1188.36/298.11 None of the processors succeeded. 1188.36/298.11 1188.36/298.11 Details of failed attempt(s): 1188.36/298.11 ----------------------------- 1188.36/298.11 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 1188.36/298.11 failed due to the following reason: 1188.36/298.11 1188.36/298.11 match-boundness of the problem could not be verified. 1188.36/298.11 1188.36/298.11 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 1188.36/298.11 failed due to the following reason: 1188.36/298.11 1188.36/298.11 match-boundness of the problem could not be verified. 1188.36/298.11 1188.36/298.11 1188.36/298.11 1188.36/298.11 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 1188.36/298.11 the following reason: 1188.36/298.11 1188.36/298.11 We add the following weak dependency pairs: 1188.36/298.11 1188.36/298.11 Strict DPs: 1188.36/298.11 { a__primes^#() -> c_1(a__sieve^#(a__from(s(s(0()))))) 1188.36/298.11 , a__primes^#() -> c_2() 1188.36/298.11 , a__sieve^#(X) -> c_3(X) 1188.36/298.11 , a__sieve^#(cons(X, Y)) -> c_4(mark^#(X), X, Y) 1188.36/298.11 , mark^#(s(X)) -> c_7(mark^#(X)) 1188.36/298.11 , mark^#(0()) -> c_8() 1188.36/298.11 , mark^#(cons(X1, X2)) -> c_9(mark^#(X1), X2) 1188.36/298.11 , mark^#(from(X)) -> c_10(a__from^#(mark(X))) 1188.36/298.11 , mark^#(true()) -> c_11() 1188.36/298.11 , mark^#(false()) -> c_12() 1188.36/298.11 , mark^#(divides(X1, X2)) -> c_13(mark^#(X1), mark^#(X2)) 1188.36/298.11 , mark^#(filter(X1, X2)) -> c_14(a__filter^#(mark(X1), mark(X2))) 1188.36/298.11 , mark^#(sieve(X)) -> c_15(a__sieve^#(mark(X))) 1188.36/298.11 , mark^#(primes()) -> c_16(a__primes^#()) 1188.36/298.11 , mark^#(head(X)) -> c_17(a__head^#(mark(X))) 1188.36/298.11 , mark^#(tail(X)) -> c_18(a__tail^#(mark(X))) 1188.36/298.11 , mark^#(if(X1, X2, X3)) -> c_19(a__if^#(mark(X1), X2, X3)) 1188.36/298.11 , a__from^#(X) -> c_5(mark^#(X), X) 1188.36/298.11 , a__from^#(X) -> c_6(X) 1188.36/298.11 , a__filter^#(X1, X2) -> c_27(X1, X2) 1188.36/298.11 , a__filter^#(s(s(X)), cons(Y, Z)) -> 1188.36/298.11 c_28(a__if^#(divides(s(s(mark(X))), mark(Y)), 1188.36/298.11 filter(s(s(X)), Z), 1188.36/298.11 cons(Y, filter(X, sieve(Y))))) 1188.36/298.11 , a__head^#(X) -> c_20(X) 1188.36/298.11 , a__head^#(cons(X, Y)) -> c_21(mark^#(X)) 1188.36/298.11 , a__tail^#(X) -> c_22(X) 1188.36/298.11 , a__tail^#(cons(X, Y)) -> c_23(mark^#(Y)) 1188.36/298.11 , a__if^#(X1, X2, X3) -> c_24(X1, X2, X3) 1188.36/298.11 , a__if^#(true(), X, Y) -> c_25(mark^#(X)) 1188.36/298.11 , a__if^#(false(), X, Y) -> c_26(mark^#(Y)) } 1188.36/298.11 1188.36/298.11 and mark the set of starting terms. 1188.36/298.11 1188.36/298.11 We are left with following problem, upon which TcT provides the 1188.36/298.11 certificate MAYBE. 1188.36/298.11 1188.36/298.11 Strict DPs: 1188.36/298.11 { a__primes^#() -> c_1(a__sieve^#(a__from(s(s(0()))))) 1188.36/298.11 , a__primes^#() -> c_2() 1188.36/298.11 , a__sieve^#(X) -> c_3(X) 1188.36/298.11 , a__sieve^#(cons(X, Y)) -> c_4(mark^#(X), X, Y) 1188.36/298.11 , mark^#(s(X)) -> c_7(mark^#(X)) 1188.36/298.11 , mark^#(0()) -> c_8() 1188.36/298.11 , mark^#(cons(X1, X2)) -> c_9(mark^#(X1), X2) 1188.36/298.11 , mark^#(from(X)) -> c_10(a__from^#(mark(X))) 1188.36/298.11 , mark^#(true()) -> c_11() 1188.36/298.11 , mark^#(false()) -> c_12() 1188.36/298.11 , mark^#(divides(X1, X2)) -> c_13(mark^#(X1), mark^#(X2)) 1188.36/298.11 , mark^#(filter(X1, X2)) -> c_14(a__filter^#(mark(X1), mark(X2))) 1188.36/298.11 , mark^#(sieve(X)) -> c_15(a__sieve^#(mark(X))) 1188.36/298.11 , mark^#(primes()) -> c_16(a__primes^#()) 1188.36/298.11 , mark^#(head(X)) -> c_17(a__head^#(mark(X))) 1188.36/298.11 , mark^#(tail(X)) -> c_18(a__tail^#(mark(X))) 1188.36/298.11 , mark^#(if(X1, X2, X3)) -> c_19(a__if^#(mark(X1), X2, X3)) 1188.36/298.11 , a__from^#(X) -> c_5(mark^#(X), X) 1188.36/298.11 , a__from^#(X) -> c_6(X) 1188.36/298.11 , a__filter^#(X1, X2) -> c_27(X1, X2) 1188.36/298.11 , a__filter^#(s(s(X)), cons(Y, Z)) -> 1188.36/298.11 c_28(a__if^#(divides(s(s(mark(X))), mark(Y)), 1188.36/298.11 filter(s(s(X)), Z), 1188.36/298.11 cons(Y, filter(X, sieve(Y))))) 1188.36/298.11 , a__head^#(X) -> c_20(X) 1188.36/298.11 , a__head^#(cons(X, Y)) -> c_21(mark^#(X)) 1188.36/298.11 , a__tail^#(X) -> c_22(X) 1188.36/298.11 , a__tail^#(cons(X, Y)) -> c_23(mark^#(Y)) 1188.36/298.11 , a__if^#(X1, X2, X3) -> c_24(X1, X2, X3) 1188.36/298.11 , a__if^#(true(), X, Y) -> c_25(mark^#(X)) 1188.36/298.11 , a__if^#(false(), X, Y) -> c_26(mark^#(Y)) } 1188.36/298.11 Strict Trs: 1188.36/298.11 { a__primes() -> a__sieve(a__from(s(s(0())))) 1188.36/298.11 , a__primes() -> primes() 1188.36/298.11 , a__sieve(X) -> sieve(X) 1188.36/298.11 , a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) 1188.36/298.11 , a__from(X) -> cons(mark(X), from(s(X))) 1188.36/298.11 , a__from(X) -> from(X) 1188.36/298.11 , mark(s(X)) -> s(mark(X)) 1188.36/298.11 , mark(0()) -> 0() 1188.36/298.11 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 1188.36/298.11 , mark(from(X)) -> a__from(mark(X)) 1188.36/298.11 , mark(true()) -> true() 1188.36/298.11 , mark(false()) -> false() 1188.36/298.11 , mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) 1188.36/298.11 , mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) 1188.36/298.11 , mark(sieve(X)) -> a__sieve(mark(X)) 1188.36/298.11 , mark(primes()) -> a__primes() 1188.36/298.11 , mark(head(X)) -> a__head(mark(X)) 1188.36/298.11 , mark(tail(X)) -> a__tail(mark(X)) 1188.36/298.11 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1188.36/298.11 , a__head(X) -> head(X) 1188.36/298.11 , a__head(cons(X, Y)) -> mark(X) 1188.36/298.11 , a__tail(X) -> tail(X) 1188.36/298.11 , a__tail(cons(X, Y)) -> mark(Y) 1188.36/298.11 , a__if(X1, X2, X3) -> if(X1, X2, X3) 1188.36/298.11 , a__if(true(), X, Y) -> mark(X) 1188.36/298.11 , a__if(false(), X, Y) -> mark(Y) 1188.36/298.11 , a__filter(X1, X2) -> filter(X1, X2) 1188.36/298.11 , a__filter(s(s(X)), cons(Y, Z)) -> 1188.36/298.11 a__if(divides(s(s(mark(X))), mark(Y)), 1188.36/298.11 filter(s(s(X)), Z), 1188.36/298.11 cons(Y, filter(X, sieve(Y)))) } 1188.36/298.11 Obligation: 1188.36/298.11 runtime complexity 1188.36/298.11 Answer: 1188.36/298.11 MAYBE 1188.36/298.11 1188.36/298.11 We estimate the number of application of {2,6,9,10} by applications 1188.36/298.11 of Pre({2,6,9,10}) = {3,4,5,7,11,14,18,19,20,22,23,24,25,26,27,28}. 1188.36/298.11 Here rules are labeled as follows: 1188.36/298.11 1188.36/298.11 DPs: 1188.36/298.11 { 1: a__primes^#() -> c_1(a__sieve^#(a__from(s(s(0()))))) 1188.36/298.11 , 2: a__primes^#() -> c_2() 1188.36/298.11 , 3: a__sieve^#(X) -> c_3(X) 1188.36/298.11 , 4: a__sieve^#(cons(X, Y)) -> c_4(mark^#(X), X, Y) 1188.36/298.11 , 5: mark^#(s(X)) -> c_7(mark^#(X)) 1188.36/298.11 , 6: mark^#(0()) -> c_8() 1188.36/298.11 , 7: mark^#(cons(X1, X2)) -> c_9(mark^#(X1), X2) 1188.36/298.11 , 8: mark^#(from(X)) -> c_10(a__from^#(mark(X))) 1188.36/298.11 , 9: mark^#(true()) -> c_11() 1188.36/298.11 , 10: mark^#(false()) -> c_12() 1188.36/298.11 , 11: mark^#(divides(X1, X2)) -> c_13(mark^#(X1), mark^#(X2)) 1188.36/298.11 , 12: mark^#(filter(X1, X2)) -> 1188.36/298.11 c_14(a__filter^#(mark(X1), mark(X2))) 1188.36/298.11 , 13: mark^#(sieve(X)) -> c_15(a__sieve^#(mark(X))) 1188.36/298.11 , 14: mark^#(primes()) -> c_16(a__primes^#()) 1188.36/298.11 , 15: mark^#(head(X)) -> c_17(a__head^#(mark(X))) 1188.36/298.11 , 16: mark^#(tail(X)) -> c_18(a__tail^#(mark(X))) 1188.36/298.11 , 17: mark^#(if(X1, X2, X3)) -> c_19(a__if^#(mark(X1), X2, X3)) 1188.36/298.11 , 18: a__from^#(X) -> c_5(mark^#(X), X) 1188.36/298.11 , 19: a__from^#(X) -> c_6(X) 1188.36/298.11 , 20: a__filter^#(X1, X2) -> c_27(X1, X2) 1188.36/298.11 , 21: a__filter^#(s(s(X)), cons(Y, Z)) -> 1188.36/298.11 c_28(a__if^#(divides(s(s(mark(X))), mark(Y)), 1188.36/298.11 filter(s(s(X)), Z), 1188.36/298.11 cons(Y, filter(X, sieve(Y))))) 1188.36/298.11 , 22: a__head^#(X) -> c_20(X) 1188.36/298.11 , 23: a__head^#(cons(X, Y)) -> c_21(mark^#(X)) 1188.36/298.11 , 24: a__tail^#(X) -> c_22(X) 1188.36/298.11 , 25: a__tail^#(cons(X, Y)) -> c_23(mark^#(Y)) 1188.36/298.11 , 26: a__if^#(X1, X2, X3) -> c_24(X1, X2, X3) 1188.36/298.11 , 27: a__if^#(true(), X, Y) -> c_25(mark^#(X)) 1188.36/298.11 , 28: a__if^#(false(), X, Y) -> c_26(mark^#(Y)) } 1188.36/298.11 1188.36/298.11 We are left with following problem, upon which TcT provides the 1188.36/298.11 certificate MAYBE. 1188.36/298.11 1188.36/298.11 Strict DPs: 1188.36/298.11 { a__primes^#() -> c_1(a__sieve^#(a__from(s(s(0()))))) 1188.36/298.11 , a__sieve^#(X) -> c_3(X) 1188.36/298.11 , a__sieve^#(cons(X, Y)) -> c_4(mark^#(X), X, Y) 1188.36/298.11 , mark^#(s(X)) -> c_7(mark^#(X)) 1188.36/298.11 , mark^#(cons(X1, X2)) -> c_9(mark^#(X1), X2) 1188.36/298.11 , mark^#(from(X)) -> c_10(a__from^#(mark(X))) 1188.36/298.11 , mark^#(divides(X1, X2)) -> c_13(mark^#(X1), mark^#(X2)) 1188.36/298.11 , mark^#(filter(X1, X2)) -> c_14(a__filter^#(mark(X1), mark(X2))) 1188.36/298.11 , mark^#(sieve(X)) -> c_15(a__sieve^#(mark(X))) 1188.36/298.11 , mark^#(primes()) -> c_16(a__primes^#()) 1188.36/298.11 , mark^#(head(X)) -> c_17(a__head^#(mark(X))) 1188.36/298.11 , mark^#(tail(X)) -> c_18(a__tail^#(mark(X))) 1188.36/298.11 , mark^#(if(X1, X2, X3)) -> c_19(a__if^#(mark(X1), X2, X3)) 1188.36/298.11 , a__from^#(X) -> c_5(mark^#(X), X) 1188.36/298.11 , a__from^#(X) -> c_6(X) 1188.36/298.11 , a__filter^#(X1, X2) -> c_27(X1, X2) 1188.36/298.11 , a__filter^#(s(s(X)), cons(Y, Z)) -> 1188.36/298.11 c_28(a__if^#(divides(s(s(mark(X))), mark(Y)), 1188.36/298.11 filter(s(s(X)), Z), 1188.36/298.11 cons(Y, filter(X, sieve(Y))))) 1188.36/298.11 , a__head^#(X) -> c_20(X) 1188.36/298.11 , a__head^#(cons(X, Y)) -> c_21(mark^#(X)) 1188.36/298.11 , a__tail^#(X) -> c_22(X) 1188.36/298.11 , a__tail^#(cons(X, Y)) -> c_23(mark^#(Y)) 1188.36/298.11 , a__if^#(X1, X2, X3) -> c_24(X1, X2, X3) 1188.36/298.11 , a__if^#(true(), X, Y) -> c_25(mark^#(X)) 1188.36/298.11 , a__if^#(false(), X, Y) -> c_26(mark^#(Y)) } 1188.36/298.11 Strict Trs: 1188.36/298.11 { a__primes() -> a__sieve(a__from(s(s(0())))) 1188.36/298.11 , a__primes() -> primes() 1188.36/298.11 , a__sieve(X) -> sieve(X) 1188.36/298.11 , a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) 1188.36/298.11 , a__from(X) -> cons(mark(X), from(s(X))) 1188.36/298.11 , a__from(X) -> from(X) 1188.36/298.11 , mark(s(X)) -> s(mark(X)) 1188.36/298.11 , mark(0()) -> 0() 1188.36/298.11 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 1188.36/298.11 , mark(from(X)) -> a__from(mark(X)) 1188.36/298.11 , mark(true()) -> true() 1188.36/298.11 , mark(false()) -> false() 1188.36/298.11 , mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) 1188.36/298.11 , mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) 1188.36/298.11 , mark(sieve(X)) -> a__sieve(mark(X)) 1188.36/298.11 , mark(primes()) -> a__primes() 1188.36/298.11 , mark(head(X)) -> a__head(mark(X)) 1188.36/298.11 , mark(tail(X)) -> a__tail(mark(X)) 1188.36/298.11 , mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) 1188.36/298.11 , a__head(X) -> head(X) 1188.36/298.11 , a__head(cons(X, Y)) -> mark(X) 1188.36/298.11 , a__tail(X) -> tail(X) 1188.36/298.11 , a__tail(cons(X, Y)) -> mark(Y) 1188.36/298.11 , a__if(X1, X2, X3) -> if(X1, X2, X3) 1188.36/298.11 , a__if(true(), X, Y) -> mark(X) 1188.36/298.11 , a__if(false(), X, Y) -> mark(Y) 1188.36/298.11 , a__filter(X1, X2) -> filter(X1, X2) 1188.36/298.11 , a__filter(s(s(X)), cons(Y, Z)) -> 1188.36/298.11 a__if(divides(s(s(mark(X))), mark(Y)), 1188.36/298.11 filter(s(s(X)), Z), 1188.36/298.11 cons(Y, filter(X, sieve(Y)))) } 1188.36/298.11 Weak DPs: 1188.36/298.11 { a__primes^#() -> c_2() 1188.36/298.11 , mark^#(0()) -> c_8() 1188.36/298.11 , mark^#(true()) -> c_11() 1188.36/298.11 , mark^#(false()) -> c_12() } 1188.36/298.11 Obligation: 1188.36/298.11 runtime complexity 1188.36/298.11 Answer: 1188.36/298.11 MAYBE 1188.36/298.11 1188.36/298.11 Empty strict component of the problem is NOT empty. 1188.36/298.11 1188.36/298.11 1188.36/298.11 Arrrr.. 1189.05/298.85 EOF