MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a__fst(X1, X2) -> fst(X1, X2) , a__fst(0(), Z) -> nil() , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) , mark(0()) -> 0() , mark(nil()) -> nil() , mark(s(X)) -> s(X) , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) , mark(from(X)) -> a__from(mark(X)) , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) , mark(len(X)) -> a__len(mark(X)) , a__from(X) -> cons(mark(X), from(s(X))) , a__from(X) -> from(X) , a__add(X1, X2) -> add(X1, X2) , a__add(0(), X) -> mark(X) , a__add(s(X), Y) -> s(add(X, Y)) , a__len(X) -> len(X) , a__len(nil()) -> 0() , a__len(cons(X, Z)) -> s(len(Z)) } 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: { a__fst^#(X1, X2) -> c_1(X1, X2) , a__fst^#(0(), Z) -> c_2() , a__fst^#(s(X), cons(Y, Z)) -> c_3(mark^#(Y), X, Z) , mark^#(0()) -> c_4() , mark^#(nil()) -> c_5() , mark^#(s(X)) -> c_6(X) , mark^#(cons(X1, X2)) -> c_7(mark^#(X1), X2) , mark^#(fst(X1, X2)) -> c_8(a__fst^#(mark(X1), mark(X2))) , mark^#(from(X)) -> c_9(a__from^#(mark(X))) , mark^#(add(X1, X2)) -> c_10(a__add^#(mark(X1), mark(X2))) , mark^#(len(X)) -> c_11(a__len^#(mark(X))) , a__from^#(X) -> c_12(mark^#(X), X) , a__from^#(X) -> c_13(X) , a__add^#(X1, X2) -> c_14(X1, X2) , a__add^#(0(), X) -> c_15(mark^#(X)) , a__add^#(s(X), Y) -> c_16(X, Y) , a__len^#(X) -> c_17(X) , a__len^#(nil()) -> c_18() , a__len^#(cons(X, Z)) -> c_19(Z) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__fst^#(X1, X2) -> c_1(X1, X2) , a__fst^#(0(), Z) -> c_2() , a__fst^#(s(X), cons(Y, Z)) -> c_3(mark^#(Y), X, Z) , mark^#(0()) -> c_4() , mark^#(nil()) -> c_5() , mark^#(s(X)) -> c_6(X) , mark^#(cons(X1, X2)) -> c_7(mark^#(X1), X2) , mark^#(fst(X1, X2)) -> c_8(a__fst^#(mark(X1), mark(X2))) , mark^#(from(X)) -> c_9(a__from^#(mark(X))) , mark^#(add(X1, X2)) -> c_10(a__add^#(mark(X1), mark(X2))) , mark^#(len(X)) -> c_11(a__len^#(mark(X))) , a__from^#(X) -> c_12(mark^#(X), X) , a__from^#(X) -> c_13(X) , a__add^#(X1, X2) -> c_14(X1, X2) , a__add^#(0(), X) -> c_15(mark^#(X)) , a__add^#(s(X), Y) -> c_16(X, Y) , a__len^#(X) -> c_17(X) , a__len^#(nil()) -> c_18() , a__len^#(cons(X, Z)) -> c_19(Z) } Strict Trs: { a__fst(X1, X2) -> fst(X1, X2) , a__fst(0(), Z) -> nil() , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) , mark(0()) -> 0() , mark(nil()) -> nil() , mark(s(X)) -> s(X) , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) , mark(from(X)) -> a__from(mark(X)) , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) , mark(len(X)) -> a__len(mark(X)) , a__from(X) -> cons(mark(X), from(s(X))) , a__from(X) -> from(X) , a__add(X1, X2) -> add(X1, X2) , a__add(0(), X) -> mark(X) , a__add(s(X), Y) -> s(add(X, Y)) , a__len(X) -> len(X) , a__len(nil()) -> 0() , a__len(cons(X, Z)) -> s(len(Z)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,4,5,18} by applications of Pre({2,4,5,18}) = {1,3,6,7,8,11,12,13,14,15,16,17,19}. Here rules are labeled as follows: DPs: { 1: a__fst^#(X1, X2) -> c_1(X1, X2) , 2: a__fst^#(0(), Z) -> c_2() , 3: a__fst^#(s(X), cons(Y, Z)) -> c_3(mark^#(Y), X, Z) , 4: mark^#(0()) -> c_4() , 5: mark^#(nil()) -> c_5() , 6: mark^#(s(X)) -> c_6(X) , 7: mark^#(cons(X1, X2)) -> c_7(mark^#(X1), X2) , 8: mark^#(fst(X1, X2)) -> c_8(a__fst^#(mark(X1), mark(X2))) , 9: mark^#(from(X)) -> c_9(a__from^#(mark(X))) , 10: mark^#(add(X1, X2)) -> c_10(a__add^#(mark(X1), mark(X2))) , 11: mark^#(len(X)) -> c_11(a__len^#(mark(X))) , 12: a__from^#(X) -> c_12(mark^#(X), X) , 13: a__from^#(X) -> c_13(X) , 14: a__add^#(X1, X2) -> c_14(X1, X2) , 15: a__add^#(0(), X) -> c_15(mark^#(X)) , 16: a__add^#(s(X), Y) -> c_16(X, Y) , 17: a__len^#(X) -> c_17(X) , 18: a__len^#(nil()) -> c_18() , 19: a__len^#(cons(X, Z)) -> c_19(Z) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__fst^#(X1, X2) -> c_1(X1, X2) , a__fst^#(s(X), cons(Y, Z)) -> c_3(mark^#(Y), X, Z) , mark^#(s(X)) -> c_6(X) , mark^#(cons(X1, X2)) -> c_7(mark^#(X1), X2) , mark^#(fst(X1, X2)) -> c_8(a__fst^#(mark(X1), mark(X2))) , mark^#(from(X)) -> c_9(a__from^#(mark(X))) , mark^#(add(X1, X2)) -> c_10(a__add^#(mark(X1), mark(X2))) , mark^#(len(X)) -> c_11(a__len^#(mark(X))) , a__from^#(X) -> c_12(mark^#(X), X) , a__from^#(X) -> c_13(X) , a__add^#(X1, X2) -> c_14(X1, X2) , a__add^#(0(), X) -> c_15(mark^#(X)) , a__add^#(s(X), Y) -> c_16(X, Y) , a__len^#(X) -> c_17(X) , a__len^#(cons(X, Z)) -> c_19(Z) } Strict Trs: { a__fst(X1, X2) -> fst(X1, X2) , a__fst(0(), Z) -> nil() , a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z)) , mark(0()) -> 0() , mark(nil()) -> nil() , mark(s(X)) -> s(X) , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2)) , mark(from(X)) -> a__from(mark(X)) , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) , mark(len(X)) -> a__len(mark(X)) , a__from(X) -> cons(mark(X), from(s(X))) , a__from(X) -> from(X) , a__add(X1, X2) -> add(X1, X2) , a__add(0(), X) -> mark(X) , a__add(s(X), Y) -> s(add(X, Y)) , a__len(X) -> len(X) , a__len(nil()) -> 0() , a__len(cons(X, Z)) -> s(len(Z)) } Weak DPs: { a__fst^#(0(), Z) -> c_2() , mark^#(0()) -> c_4() , mark^#(nil()) -> c_5() , a__len^#(nil()) -> c_18() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..