MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a__from(X) -> cons(mark(X), from(s(X))) , a__from(X) -> from(X) , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(from(X)) -> a__from(mark(X)) , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(rnil()) -> rnil() , mark(cons2(X1, X2)) -> cons2(X1, mark(X2)) , mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2)) , mark(posrecip(X)) -> posrecip(mark(X)) , mark(negrecip(X)) -> negrecip(mark(X)) , mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2)) , mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2)) , mark(pi(X)) -> a__pi(mark(X)) , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) , mark(times(X1, X2)) -> a__times(mark(X1), mark(X2)) , mark(square(X)) -> a__square(mark(X)) , mark(nil()) -> nil() , a__2ndspos(X1, X2) -> 2ndspos(X1, X2) , a__2ndspos(s(N), cons(X, Z)) -> a__2ndspos(s(mark(N)), cons2(X, mark(Z))) , a__2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z))) , a__2ndspos(0(), Z) -> rnil() , a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2) , a__2ndsneg(s(N), cons(X, Z)) -> a__2ndsneg(s(mark(N)), cons2(X, mark(Z))) , a__2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z))) , a__2ndsneg(0(), Z) -> rnil() , a__pi(X) -> a__2ndspos(mark(X), a__from(0())) , a__pi(X) -> pi(X) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y))) , a__plus(0(), Y) -> mark(Y) , a__times(X1, X2) -> times(X1, X2) , a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y))) , a__times(0(), Y) -> 0() , a__square(X) -> a__times(mark(X), mark(X)) , a__square(X) -> square(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: { a__from^#(X) -> c_1(mark^#(X), X) , a__from^#(X) -> c_2(X) , mark^#(cons(X1, X2)) -> c_3(mark^#(X1), X2) , mark^#(from(X)) -> c_4(a__from^#(mark(X))) , mark^#(s(X)) -> c_5(mark^#(X)) , mark^#(0()) -> c_6() , mark^#(rnil()) -> c_7() , mark^#(cons2(X1, X2)) -> c_8(X1, mark^#(X2)) , mark^#(rcons(X1, X2)) -> c_9(mark^#(X1), mark^#(X2)) , mark^#(posrecip(X)) -> c_10(mark^#(X)) , mark^#(negrecip(X)) -> c_11(mark^#(X)) , mark^#(2ndspos(X1, X2)) -> c_12(a__2ndspos^#(mark(X1), mark(X2))) , mark^#(2ndsneg(X1, X2)) -> c_13(a__2ndsneg^#(mark(X1), mark(X2))) , mark^#(pi(X)) -> c_14(a__pi^#(mark(X))) , mark^#(plus(X1, X2)) -> c_15(a__plus^#(mark(X1), mark(X2))) , mark^#(times(X1, X2)) -> c_16(a__times^#(mark(X1), mark(X2))) , mark^#(square(X)) -> c_17(a__square^#(mark(X))) , mark^#(nil()) -> c_18() , a__2ndspos^#(X1, X2) -> c_19(X1, X2) , a__2ndspos^#(s(N), cons(X, Z)) -> c_20(a__2ndspos^#(s(mark(N)), cons2(X, mark(Z)))) , a__2ndspos^#(s(N), cons2(X, cons(Y, Z))) -> c_21(mark^#(Y), a__2ndsneg^#(mark(N), mark(Z))) , a__2ndspos^#(0(), Z) -> c_22() , a__2ndsneg^#(X1, X2) -> c_23(X1, X2) , a__2ndsneg^#(s(N), cons(X, Z)) -> c_24(a__2ndsneg^#(s(mark(N)), cons2(X, mark(Z)))) , a__2ndsneg^#(s(N), cons2(X, cons(Y, Z))) -> c_25(mark^#(Y), a__2ndspos^#(mark(N), mark(Z))) , a__2ndsneg^#(0(), Z) -> c_26() , a__pi^#(X) -> c_27(a__2ndspos^#(mark(X), a__from(0()))) , a__pi^#(X) -> c_28(X) , a__plus^#(X1, X2) -> c_29(X1, X2) , a__plus^#(s(X), Y) -> c_30(a__plus^#(mark(X), mark(Y))) , a__plus^#(0(), Y) -> c_31(mark^#(Y)) , a__times^#(X1, X2) -> c_32(X1, X2) , a__times^#(s(X), Y) -> c_33(a__plus^#(mark(Y), a__times(mark(X), mark(Y)))) , a__times^#(0(), Y) -> c_34() , a__square^#(X) -> c_35(a__times^#(mark(X), mark(X))) , a__square^#(X) -> c_36(X) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__from^#(X) -> c_1(mark^#(X), X) , a__from^#(X) -> c_2(X) , mark^#(cons(X1, X2)) -> c_3(mark^#(X1), X2) , mark^#(from(X)) -> c_4(a__from^#(mark(X))) , mark^#(s(X)) -> c_5(mark^#(X)) , mark^#(0()) -> c_6() , mark^#(rnil()) -> c_7() , mark^#(cons2(X1, X2)) -> c_8(X1, mark^#(X2)) , mark^#(rcons(X1, X2)) -> c_9(mark^#(X1), mark^#(X2)) , mark^#(posrecip(X)) -> c_10(mark^#(X)) , mark^#(negrecip(X)) -> c_11(mark^#(X)) , mark^#(2ndspos(X1, X2)) -> c_12(a__2ndspos^#(mark(X1), mark(X2))) , mark^#(2ndsneg(X1, X2)) -> c_13(a__2ndsneg^#(mark(X1), mark(X2))) , mark^#(pi(X)) -> c_14(a__pi^#(mark(X))) , mark^#(plus(X1, X2)) -> c_15(a__plus^#(mark(X1), mark(X2))) , mark^#(times(X1, X2)) -> c_16(a__times^#(mark(X1), mark(X2))) , mark^#(square(X)) -> c_17(a__square^#(mark(X))) , mark^#(nil()) -> c_18() , a__2ndspos^#(X1, X2) -> c_19(X1, X2) , a__2ndspos^#(s(N), cons(X, Z)) -> c_20(a__2ndspos^#(s(mark(N)), cons2(X, mark(Z)))) , a__2ndspos^#(s(N), cons2(X, cons(Y, Z))) -> c_21(mark^#(Y), a__2ndsneg^#(mark(N), mark(Z))) , a__2ndspos^#(0(), Z) -> c_22() , a__2ndsneg^#(X1, X2) -> c_23(X1, X2) , a__2ndsneg^#(s(N), cons(X, Z)) -> c_24(a__2ndsneg^#(s(mark(N)), cons2(X, mark(Z)))) , a__2ndsneg^#(s(N), cons2(X, cons(Y, Z))) -> c_25(mark^#(Y), a__2ndspos^#(mark(N), mark(Z))) , a__2ndsneg^#(0(), Z) -> c_26() , a__pi^#(X) -> c_27(a__2ndspos^#(mark(X), a__from(0()))) , a__pi^#(X) -> c_28(X) , a__plus^#(X1, X2) -> c_29(X1, X2) , a__plus^#(s(X), Y) -> c_30(a__plus^#(mark(X), mark(Y))) , a__plus^#(0(), Y) -> c_31(mark^#(Y)) , a__times^#(X1, X2) -> c_32(X1, X2) , a__times^#(s(X), Y) -> c_33(a__plus^#(mark(Y), a__times(mark(X), mark(Y)))) , a__times^#(0(), Y) -> c_34() , a__square^#(X) -> c_35(a__times^#(mark(X), mark(X))) , a__square^#(X) -> c_36(X) } Strict Trs: { a__from(X) -> cons(mark(X), from(s(X))) , a__from(X) -> from(X) , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(from(X)) -> a__from(mark(X)) , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(rnil()) -> rnil() , mark(cons2(X1, X2)) -> cons2(X1, mark(X2)) , mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2)) , mark(posrecip(X)) -> posrecip(mark(X)) , mark(negrecip(X)) -> negrecip(mark(X)) , mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2)) , mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2)) , mark(pi(X)) -> a__pi(mark(X)) , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) , mark(times(X1, X2)) -> a__times(mark(X1), mark(X2)) , mark(square(X)) -> a__square(mark(X)) , mark(nil()) -> nil() , a__2ndspos(X1, X2) -> 2ndspos(X1, X2) , a__2ndspos(s(N), cons(X, Z)) -> a__2ndspos(s(mark(N)), cons2(X, mark(Z))) , a__2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z))) , a__2ndspos(0(), Z) -> rnil() , a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2) , a__2ndsneg(s(N), cons(X, Z)) -> a__2ndsneg(s(mark(N)), cons2(X, mark(Z))) , a__2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z))) , a__2ndsneg(0(), Z) -> rnil() , a__pi(X) -> a__2ndspos(mark(X), a__from(0())) , a__pi(X) -> pi(X) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y))) , a__plus(0(), Y) -> mark(Y) , a__times(X1, X2) -> times(X1, X2) , a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y))) , a__times(0(), Y) -> 0() , a__square(X) -> a__times(mark(X), mark(X)) , a__square(X) -> square(X) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {6,7,18,22,26,34} by applications of Pre({6,7,18,22,26,34}) = {1,2,3,5,8,9,10,11,12,13,16,19,21,23,25,27,28,29,31,32,35,36}. Here rules are labeled as follows: DPs: { 1: a__from^#(X) -> c_1(mark^#(X), X) , 2: a__from^#(X) -> c_2(X) , 3: mark^#(cons(X1, X2)) -> c_3(mark^#(X1), X2) , 4: mark^#(from(X)) -> c_4(a__from^#(mark(X))) , 5: mark^#(s(X)) -> c_5(mark^#(X)) , 6: mark^#(0()) -> c_6() , 7: mark^#(rnil()) -> c_7() , 8: mark^#(cons2(X1, X2)) -> c_8(X1, mark^#(X2)) , 9: mark^#(rcons(X1, X2)) -> c_9(mark^#(X1), mark^#(X2)) , 10: mark^#(posrecip(X)) -> c_10(mark^#(X)) , 11: mark^#(negrecip(X)) -> c_11(mark^#(X)) , 12: mark^#(2ndspos(X1, X2)) -> c_12(a__2ndspos^#(mark(X1), mark(X2))) , 13: mark^#(2ndsneg(X1, X2)) -> c_13(a__2ndsneg^#(mark(X1), mark(X2))) , 14: mark^#(pi(X)) -> c_14(a__pi^#(mark(X))) , 15: mark^#(plus(X1, X2)) -> c_15(a__plus^#(mark(X1), mark(X2))) , 16: mark^#(times(X1, X2)) -> c_16(a__times^#(mark(X1), mark(X2))) , 17: mark^#(square(X)) -> c_17(a__square^#(mark(X))) , 18: mark^#(nil()) -> c_18() , 19: a__2ndspos^#(X1, X2) -> c_19(X1, X2) , 20: a__2ndspos^#(s(N), cons(X, Z)) -> c_20(a__2ndspos^#(s(mark(N)), cons2(X, mark(Z)))) , 21: a__2ndspos^#(s(N), cons2(X, cons(Y, Z))) -> c_21(mark^#(Y), a__2ndsneg^#(mark(N), mark(Z))) , 22: a__2ndspos^#(0(), Z) -> c_22() , 23: a__2ndsneg^#(X1, X2) -> c_23(X1, X2) , 24: a__2ndsneg^#(s(N), cons(X, Z)) -> c_24(a__2ndsneg^#(s(mark(N)), cons2(X, mark(Z)))) , 25: a__2ndsneg^#(s(N), cons2(X, cons(Y, Z))) -> c_25(mark^#(Y), a__2ndspos^#(mark(N), mark(Z))) , 26: a__2ndsneg^#(0(), Z) -> c_26() , 27: a__pi^#(X) -> c_27(a__2ndspos^#(mark(X), a__from(0()))) , 28: a__pi^#(X) -> c_28(X) , 29: a__plus^#(X1, X2) -> c_29(X1, X2) , 30: a__plus^#(s(X), Y) -> c_30(a__plus^#(mark(X), mark(Y))) , 31: a__plus^#(0(), Y) -> c_31(mark^#(Y)) , 32: a__times^#(X1, X2) -> c_32(X1, X2) , 33: a__times^#(s(X), Y) -> c_33(a__plus^#(mark(Y), a__times(mark(X), mark(Y)))) , 34: a__times^#(0(), Y) -> c_34() , 35: a__square^#(X) -> c_35(a__times^#(mark(X), mark(X))) , 36: a__square^#(X) -> c_36(X) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__from^#(X) -> c_1(mark^#(X), X) , a__from^#(X) -> c_2(X) , mark^#(cons(X1, X2)) -> c_3(mark^#(X1), X2) , mark^#(from(X)) -> c_4(a__from^#(mark(X))) , mark^#(s(X)) -> c_5(mark^#(X)) , mark^#(cons2(X1, X2)) -> c_8(X1, mark^#(X2)) , mark^#(rcons(X1, X2)) -> c_9(mark^#(X1), mark^#(X2)) , mark^#(posrecip(X)) -> c_10(mark^#(X)) , mark^#(negrecip(X)) -> c_11(mark^#(X)) , mark^#(2ndspos(X1, X2)) -> c_12(a__2ndspos^#(mark(X1), mark(X2))) , mark^#(2ndsneg(X1, X2)) -> c_13(a__2ndsneg^#(mark(X1), mark(X2))) , mark^#(pi(X)) -> c_14(a__pi^#(mark(X))) , mark^#(plus(X1, X2)) -> c_15(a__plus^#(mark(X1), mark(X2))) , mark^#(times(X1, X2)) -> c_16(a__times^#(mark(X1), mark(X2))) , mark^#(square(X)) -> c_17(a__square^#(mark(X))) , a__2ndspos^#(X1, X2) -> c_19(X1, X2) , a__2ndspos^#(s(N), cons(X, Z)) -> c_20(a__2ndspos^#(s(mark(N)), cons2(X, mark(Z)))) , a__2ndspos^#(s(N), cons2(X, cons(Y, Z))) -> c_21(mark^#(Y), a__2ndsneg^#(mark(N), mark(Z))) , a__2ndsneg^#(X1, X2) -> c_23(X1, X2) , a__2ndsneg^#(s(N), cons(X, Z)) -> c_24(a__2ndsneg^#(s(mark(N)), cons2(X, mark(Z)))) , a__2ndsneg^#(s(N), cons2(X, cons(Y, Z))) -> c_25(mark^#(Y), a__2ndspos^#(mark(N), mark(Z))) , a__pi^#(X) -> c_27(a__2ndspos^#(mark(X), a__from(0()))) , a__pi^#(X) -> c_28(X) , a__plus^#(X1, X2) -> c_29(X1, X2) , a__plus^#(s(X), Y) -> c_30(a__plus^#(mark(X), mark(Y))) , a__plus^#(0(), Y) -> c_31(mark^#(Y)) , a__times^#(X1, X2) -> c_32(X1, X2) , a__times^#(s(X), Y) -> c_33(a__plus^#(mark(Y), a__times(mark(X), mark(Y)))) , a__square^#(X) -> c_35(a__times^#(mark(X), mark(X))) , a__square^#(X) -> c_36(X) } Strict Trs: { a__from(X) -> cons(mark(X), from(s(X))) , a__from(X) -> from(X) , mark(cons(X1, X2)) -> cons(mark(X1), X2) , mark(from(X)) -> a__from(mark(X)) , mark(s(X)) -> s(mark(X)) , mark(0()) -> 0() , mark(rnil()) -> rnil() , mark(cons2(X1, X2)) -> cons2(X1, mark(X2)) , mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2)) , mark(posrecip(X)) -> posrecip(mark(X)) , mark(negrecip(X)) -> negrecip(mark(X)) , mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2)) , mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2)) , mark(pi(X)) -> a__pi(mark(X)) , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) , mark(times(X1, X2)) -> a__times(mark(X1), mark(X2)) , mark(square(X)) -> a__square(mark(X)) , mark(nil()) -> nil() , a__2ndspos(X1, X2) -> 2ndspos(X1, X2) , a__2ndspos(s(N), cons(X, Z)) -> a__2ndspos(s(mark(N)), cons2(X, mark(Z))) , a__2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z))) , a__2ndspos(0(), Z) -> rnil() , a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2) , a__2ndsneg(s(N), cons(X, Z)) -> a__2ndsneg(s(mark(N)), cons2(X, mark(Z))) , a__2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z))) , a__2ndsneg(0(), Z) -> rnil() , a__pi(X) -> a__2ndspos(mark(X), a__from(0())) , a__pi(X) -> pi(X) , a__plus(X1, X2) -> plus(X1, X2) , a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y))) , a__plus(0(), Y) -> mark(Y) , a__times(X1, X2) -> times(X1, X2) , a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y))) , a__times(0(), Y) -> 0() , a__square(X) -> a__times(mark(X), mark(X)) , a__square(X) -> square(X) } Weak DPs: { mark^#(0()) -> c_6() , mark^#(rnil()) -> c_7() , mark^#(nil()) -> c_18() , a__2ndspos^#(0(), Z) -> c_22() , a__2ndsneg^#(0(), Z) -> c_26() , a__times^#(0(), Y) -> c_34() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..