MAYBE 1185.27/297.38 MAYBE 1185.27/297.38 1185.27/297.38 We are left with following problem, upon which TcT provides the 1185.27/297.38 certificate MAYBE. 1185.27/297.38 1185.27/297.38 Strict Trs: 1185.27/297.38 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 1185.27/297.38 , a__terms(X) -> terms(X) 1185.27/297.38 , a__sqr(X) -> sqr(X) 1185.27/297.38 , a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) 1185.27/297.38 , a__sqr(0()) -> 0() 1185.27/297.38 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 1185.27/297.38 , mark(recip(X)) -> recip(mark(X)) 1185.27/297.38 , mark(terms(X)) -> a__terms(mark(X)) 1185.27/297.38 , mark(s(X)) -> s(mark(X)) 1185.27/297.38 , mark(0()) -> 0() 1185.27/297.38 , mark(nil()) -> nil() 1185.27/297.38 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1185.27/297.38 , mark(dbl(X)) -> a__dbl(mark(X)) 1185.27/297.38 , mark(sqr(X)) -> a__sqr(mark(X)) 1185.27/297.38 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 1185.27/297.38 , mark(half(X)) -> a__half(mark(X)) 1185.27/297.38 , a__add(X1, X2) -> add(X1, X2) 1185.27/297.38 , a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) 1185.27/297.38 , a__add(0(), X) -> mark(X) 1185.27/297.38 , a__dbl(X) -> dbl(X) 1185.27/297.38 , a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) 1185.27/297.38 , a__dbl(0()) -> 0() 1185.27/297.38 , a__first(X1, X2) -> first(X1, X2) 1185.27/297.38 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1185.27/297.38 , a__first(0(), X) -> nil() 1185.27/297.38 , a__half(X) -> half(X) 1185.27/297.38 , a__half(s(s(X))) -> s(a__half(mark(X))) 1185.27/297.38 , a__half(s(0())) -> 0() 1185.27/297.38 , a__half(0()) -> 0() 1185.27/297.38 , a__half(dbl(X)) -> mark(X) } 1185.27/297.38 Obligation: 1185.27/297.38 runtime complexity 1185.27/297.38 Answer: 1185.27/297.38 MAYBE 1185.27/297.38 1185.27/297.38 None of the processors succeeded. 1185.27/297.38 1185.27/297.38 Details of failed attempt(s): 1185.27/297.38 ----------------------------- 1185.27/297.38 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 1185.27/297.38 following reason: 1185.27/297.38 1185.27/297.38 Computation stopped due to timeout after 297.0 seconds. 1185.27/297.38 1185.27/297.38 2) 'Best' failed due to the following reason: 1185.27/297.38 1185.27/297.38 None of the processors succeeded. 1185.27/297.38 1185.27/297.38 Details of failed attempt(s): 1185.27/297.38 ----------------------------- 1185.27/297.38 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 1185.27/297.38 seconds)' failed due to the following reason: 1185.27/297.38 1185.27/297.38 Computation stopped due to timeout after 148.0 seconds. 1185.27/297.38 1185.27/297.38 2) 'Best' failed due to the following reason: 1185.27/297.38 1185.27/297.38 None of the processors succeeded. 1185.27/297.38 1185.27/297.38 Details of failed attempt(s): 1185.27/297.38 ----------------------------- 1185.27/297.38 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 1185.27/297.38 following reason: 1185.27/297.38 1185.27/297.38 The processor is inapplicable, reason: 1185.27/297.38 Processor only applicable for innermost runtime complexity analysis 1185.27/297.38 1185.27/297.38 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 1185.27/297.38 to the following reason: 1185.27/297.38 1185.27/297.38 The processor is inapplicable, reason: 1185.27/297.38 Processor only applicable for innermost runtime complexity analysis 1185.27/297.38 1185.27/297.38 1185.27/297.38 3) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 1185.27/297.38 failed due to the following reason: 1185.27/297.38 1185.27/297.38 None of the processors succeeded. 1185.27/297.38 1185.27/297.38 Details of failed attempt(s): 1185.27/297.38 ----------------------------- 1185.27/297.38 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 1185.27/297.38 failed due to the following reason: 1185.27/297.38 1185.27/297.38 match-boundness of the problem could not be verified. 1185.27/297.38 1185.27/297.38 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 1185.27/297.38 failed due to the following reason: 1185.27/297.38 1185.27/297.38 match-boundness of the problem could not be verified. 1185.27/297.38 1185.27/297.38 1185.27/297.38 1185.27/297.38 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 1185.27/297.38 the following reason: 1185.27/297.38 1185.27/297.38 We add the following weak dependency pairs: 1185.27/297.38 1185.27/297.38 Strict DPs: 1185.27/297.38 { a__terms^#(N) -> c_1(a__sqr^#(mark(N)), N) 1185.27/297.38 , a__terms^#(X) -> c_2(X) 1185.27/297.38 , a__sqr^#(X) -> c_3(X) 1185.27/297.38 , a__sqr^#(s(X)) -> c_4(a__add^#(a__sqr(mark(X)), a__dbl(mark(X)))) 1185.27/297.38 , a__sqr^#(0()) -> c_5() 1185.27/297.38 , a__add^#(X1, X2) -> c_17(X1, X2) 1185.27/297.38 , a__add^#(s(X), Y) -> c_18(a__add^#(mark(X), mark(Y))) 1185.27/297.38 , a__add^#(0(), X) -> c_19(mark^#(X)) 1185.27/297.38 , mark^#(cons(X1, X2)) -> c_6(mark^#(X1), X2) 1185.27/297.38 , mark^#(recip(X)) -> c_7(mark^#(X)) 1185.27/297.38 , mark^#(terms(X)) -> c_8(a__terms^#(mark(X))) 1185.27/297.38 , mark^#(s(X)) -> c_9(mark^#(X)) 1185.27/297.38 , mark^#(0()) -> c_10() 1185.27/297.38 , mark^#(nil()) -> c_11() 1185.27/297.38 , mark^#(first(X1, X2)) -> c_12(a__first^#(mark(X1), mark(X2))) 1185.27/297.38 , mark^#(dbl(X)) -> c_13(a__dbl^#(mark(X))) 1185.27/297.38 , mark^#(sqr(X)) -> c_14(a__sqr^#(mark(X))) 1185.27/297.38 , mark^#(add(X1, X2)) -> c_15(a__add^#(mark(X1), mark(X2))) 1185.27/297.38 , mark^#(half(X)) -> c_16(a__half^#(mark(X))) 1185.27/297.38 , a__first^#(X1, X2) -> c_23(X1, X2) 1185.27/297.38 , a__first^#(s(X), cons(Y, Z)) -> c_24(mark^#(Y), X, Z) 1185.27/297.38 , a__first^#(0(), X) -> c_25() 1185.27/297.38 , a__dbl^#(X) -> c_20(X) 1185.27/297.38 , a__dbl^#(s(X)) -> c_21(a__dbl^#(mark(X))) 1185.27/297.38 , a__dbl^#(0()) -> c_22() 1185.27/297.38 , a__half^#(X) -> c_26(X) 1185.27/297.38 , a__half^#(s(s(X))) -> c_27(a__half^#(mark(X))) 1185.27/297.38 , a__half^#(s(0())) -> c_28() 1185.27/297.38 , a__half^#(0()) -> c_29() 1185.27/297.38 , a__half^#(dbl(X)) -> c_30(mark^#(X)) } 1185.27/297.38 1185.27/297.38 and mark the set of starting terms. 1185.27/297.38 1185.27/297.38 We are left with following problem, upon which TcT provides the 1185.27/297.38 certificate MAYBE. 1185.27/297.38 1185.27/297.38 Strict DPs: 1185.27/297.38 { a__terms^#(N) -> c_1(a__sqr^#(mark(N)), N) 1185.27/297.38 , a__terms^#(X) -> c_2(X) 1185.27/297.38 , a__sqr^#(X) -> c_3(X) 1185.27/297.38 , a__sqr^#(s(X)) -> c_4(a__add^#(a__sqr(mark(X)), a__dbl(mark(X)))) 1185.27/297.38 , a__sqr^#(0()) -> c_5() 1185.27/297.38 , a__add^#(X1, X2) -> c_17(X1, X2) 1185.27/297.38 , a__add^#(s(X), Y) -> c_18(a__add^#(mark(X), mark(Y))) 1185.27/297.38 , a__add^#(0(), X) -> c_19(mark^#(X)) 1185.27/297.38 , mark^#(cons(X1, X2)) -> c_6(mark^#(X1), X2) 1185.27/297.38 , mark^#(recip(X)) -> c_7(mark^#(X)) 1185.27/297.38 , mark^#(terms(X)) -> c_8(a__terms^#(mark(X))) 1185.27/297.38 , mark^#(s(X)) -> c_9(mark^#(X)) 1185.27/297.38 , mark^#(0()) -> c_10() 1185.27/297.38 , mark^#(nil()) -> c_11() 1185.27/297.38 , mark^#(first(X1, X2)) -> c_12(a__first^#(mark(X1), mark(X2))) 1185.27/297.38 , mark^#(dbl(X)) -> c_13(a__dbl^#(mark(X))) 1185.27/297.38 , mark^#(sqr(X)) -> c_14(a__sqr^#(mark(X))) 1185.27/297.38 , mark^#(add(X1, X2)) -> c_15(a__add^#(mark(X1), mark(X2))) 1185.27/297.38 , mark^#(half(X)) -> c_16(a__half^#(mark(X))) 1185.27/297.38 , a__first^#(X1, X2) -> c_23(X1, X2) 1185.27/297.38 , a__first^#(s(X), cons(Y, Z)) -> c_24(mark^#(Y), X, Z) 1185.27/297.38 , a__first^#(0(), X) -> c_25() 1185.27/297.38 , a__dbl^#(X) -> c_20(X) 1185.27/297.38 , a__dbl^#(s(X)) -> c_21(a__dbl^#(mark(X))) 1185.27/297.38 , a__dbl^#(0()) -> c_22() 1185.27/297.38 , a__half^#(X) -> c_26(X) 1185.27/297.38 , a__half^#(s(s(X))) -> c_27(a__half^#(mark(X))) 1185.27/297.38 , a__half^#(s(0())) -> c_28() 1185.27/297.38 , a__half^#(0()) -> c_29() 1185.27/297.38 , a__half^#(dbl(X)) -> c_30(mark^#(X)) } 1185.27/297.38 Strict Trs: 1185.27/297.38 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 1185.27/297.38 , a__terms(X) -> terms(X) 1185.27/297.38 , a__sqr(X) -> sqr(X) 1185.27/297.38 , a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) 1185.27/297.38 , a__sqr(0()) -> 0() 1185.27/297.38 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 1185.27/297.38 , mark(recip(X)) -> recip(mark(X)) 1185.27/297.38 , mark(terms(X)) -> a__terms(mark(X)) 1185.27/297.38 , mark(s(X)) -> s(mark(X)) 1185.27/297.38 , mark(0()) -> 0() 1185.27/297.38 , mark(nil()) -> nil() 1185.27/297.38 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1185.27/297.38 , mark(dbl(X)) -> a__dbl(mark(X)) 1185.27/297.38 , mark(sqr(X)) -> a__sqr(mark(X)) 1185.27/297.38 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 1185.27/297.38 , mark(half(X)) -> a__half(mark(X)) 1185.27/297.38 , a__add(X1, X2) -> add(X1, X2) 1185.27/297.38 , a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) 1185.27/297.38 , a__add(0(), X) -> mark(X) 1185.27/297.38 , a__dbl(X) -> dbl(X) 1185.27/297.38 , a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) 1185.27/297.38 , a__dbl(0()) -> 0() 1185.27/297.38 , a__first(X1, X2) -> first(X1, X2) 1185.27/297.38 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1185.27/297.38 , a__first(0(), X) -> nil() 1185.27/297.38 , a__half(X) -> half(X) 1185.27/297.38 , a__half(s(s(X))) -> s(a__half(mark(X))) 1185.27/297.38 , a__half(s(0())) -> 0() 1185.27/297.38 , a__half(0()) -> 0() 1185.27/297.38 , a__half(dbl(X)) -> mark(X) } 1185.27/297.38 Obligation: 1185.27/297.38 runtime complexity 1185.27/297.38 Answer: 1185.27/297.38 MAYBE 1185.27/297.38 1185.27/297.38 We estimate the number of application of {5,13,14,22,25,28,29} by 1185.27/297.38 applications of Pre({5,13,14,22,25,28,29}) = 1185.27/297.38 {1,2,3,6,8,9,10,12,15,16,17,19,20,21,23,24,26,27,30}. Here rules 1185.27/297.38 are labeled as follows: 1185.27/297.38 1185.27/297.38 DPs: 1185.27/297.38 { 1: a__terms^#(N) -> c_1(a__sqr^#(mark(N)), N) 1185.27/297.38 , 2: a__terms^#(X) -> c_2(X) 1185.27/297.38 , 3: a__sqr^#(X) -> c_3(X) 1185.27/297.38 , 4: a__sqr^#(s(X)) -> 1185.27/297.38 c_4(a__add^#(a__sqr(mark(X)), a__dbl(mark(X)))) 1185.27/297.38 , 5: a__sqr^#(0()) -> c_5() 1185.27/297.38 , 6: a__add^#(X1, X2) -> c_17(X1, X2) 1185.27/297.38 , 7: a__add^#(s(X), Y) -> c_18(a__add^#(mark(X), mark(Y))) 1185.27/297.38 , 8: a__add^#(0(), X) -> c_19(mark^#(X)) 1185.27/297.38 , 9: mark^#(cons(X1, X2)) -> c_6(mark^#(X1), X2) 1185.27/297.38 , 10: mark^#(recip(X)) -> c_7(mark^#(X)) 1185.67/297.41 , 11: mark^#(terms(X)) -> c_8(a__terms^#(mark(X))) 1185.67/297.41 , 12: mark^#(s(X)) -> c_9(mark^#(X)) 1185.67/297.41 , 13: mark^#(0()) -> c_10() 1185.67/297.41 , 14: mark^#(nil()) -> c_11() 1185.67/297.41 , 15: mark^#(first(X1, X2)) -> c_12(a__first^#(mark(X1), mark(X2))) 1185.67/297.41 , 16: mark^#(dbl(X)) -> c_13(a__dbl^#(mark(X))) 1185.67/297.41 , 17: mark^#(sqr(X)) -> c_14(a__sqr^#(mark(X))) 1185.67/297.41 , 18: mark^#(add(X1, X2)) -> c_15(a__add^#(mark(X1), mark(X2))) 1185.67/297.41 , 19: mark^#(half(X)) -> c_16(a__half^#(mark(X))) 1185.67/297.41 , 20: a__first^#(X1, X2) -> c_23(X1, X2) 1185.67/297.41 , 21: a__first^#(s(X), cons(Y, Z)) -> c_24(mark^#(Y), X, Z) 1185.67/297.41 , 22: a__first^#(0(), X) -> c_25() 1185.67/297.41 , 23: a__dbl^#(X) -> c_20(X) 1185.67/297.41 , 24: a__dbl^#(s(X)) -> c_21(a__dbl^#(mark(X))) 1185.67/297.41 , 25: a__dbl^#(0()) -> c_22() 1185.67/297.41 , 26: a__half^#(X) -> c_26(X) 1185.67/297.41 , 27: a__half^#(s(s(X))) -> c_27(a__half^#(mark(X))) 1185.67/297.41 , 28: a__half^#(s(0())) -> c_28() 1185.67/297.41 , 29: a__half^#(0()) -> c_29() 1185.67/297.41 , 30: a__half^#(dbl(X)) -> c_30(mark^#(X)) } 1185.67/297.41 1185.67/297.41 We are left with following problem, upon which TcT provides the 1185.67/297.41 certificate MAYBE. 1185.67/297.41 1185.67/297.41 Strict DPs: 1185.67/297.41 { a__terms^#(N) -> c_1(a__sqr^#(mark(N)), N) 1185.67/297.41 , a__terms^#(X) -> c_2(X) 1185.67/297.41 , a__sqr^#(X) -> c_3(X) 1185.67/297.41 , a__sqr^#(s(X)) -> c_4(a__add^#(a__sqr(mark(X)), a__dbl(mark(X)))) 1185.67/297.41 , a__add^#(X1, X2) -> c_17(X1, X2) 1185.67/297.41 , a__add^#(s(X), Y) -> c_18(a__add^#(mark(X), mark(Y))) 1185.67/297.41 , a__add^#(0(), X) -> c_19(mark^#(X)) 1185.67/297.41 , mark^#(cons(X1, X2)) -> c_6(mark^#(X1), X2) 1185.67/297.41 , mark^#(recip(X)) -> c_7(mark^#(X)) 1185.67/297.41 , mark^#(terms(X)) -> c_8(a__terms^#(mark(X))) 1185.67/297.41 , mark^#(s(X)) -> c_9(mark^#(X)) 1185.67/297.41 , mark^#(first(X1, X2)) -> c_12(a__first^#(mark(X1), mark(X2))) 1185.67/297.41 , mark^#(dbl(X)) -> c_13(a__dbl^#(mark(X))) 1185.67/297.41 , mark^#(sqr(X)) -> c_14(a__sqr^#(mark(X))) 1185.67/297.41 , mark^#(add(X1, X2)) -> c_15(a__add^#(mark(X1), mark(X2))) 1185.67/297.41 , mark^#(half(X)) -> c_16(a__half^#(mark(X))) 1185.67/297.41 , a__first^#(X1, X2) -> c_23(X1, X2) 1185.67/297.41 , a__first^#(s(X), cons(Y, Z)) -> c_24(mark^#(Y), X, Z) 1185.67/297.41 , a__dbl^#(X) -> c_20(X) 1185.67/297.41 , a__dbl^#(s(X)) -> c_21(a__dbl^#(mark(X))) 1185.67/297.41 , a__half^#(X) -> c_26(X) 1185.67/297.41 , a__half^#(s(s(X))) -> c_27(a__half^#(mark(X))) 1185.67/297.41 , a__half^#(dbl(X)) -> c_30(mark^#(X)) } 1185.67/297.41 Strict Trs: 1185.67/297.41 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 1185.67/297.41 , a__terms(X) -> terms(X) 1185.67/297.41 , a__sqr(X) -> sqr(X) 1185.67/297.41 , a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) 1185.67/297.41 , a__sqr(0()) -> 0() 1185.67/297.41 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 1185.67/297.41 , mark(recip(X)) -> recip(mark(X)) 1185.67/297.41 , mark(terms(X)) -> a__terms(mark(X)) 1185.67/297.41 , mark(s(X)) -> s(mark(X)) 1185.67/297.41 , mark(0()) -> 0() 1185.67/297.41 , mark(nil()) -> nil() 1185.67/297.41 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1185.67/297.41 , mark(dbl(X)) -> a__dbl(mark(X)) 1185.67/297.41 , mark(sqr(X)) -> a__sqr(mark(X)) 1185.67/297.41 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 1185.67/297.41 , mark(half(X)) -> a__half(mark(X)) 1185.67/297.41 , a__add(X1, X2) -> add(X1, X2) 1185.67/297.41 , a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) 1185.67/297.41 , a__add(0(), X) -> mark(X) 1185.67/297.41 , a__dbl(X) -> dbl(X) 1185.67/297.41 , a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) 1185.67/297.41 , a__dbl(0()) -> 0() 1185.67/297.41 , a__first(X1, X2) -> first(X1, X2) 1185.67/297.41 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1185.67/297.41 , a__first(0(), X) -> nil() 1185.67/297.41 , a__half(X) -> half(X) 1185.67/297.41 , a__half(s(s(X))) -> s(a__half(mark(X))) 1185.67/297.41 , a__half(s(0())) -> 0() 1185.67/297.41 , a__half(0()) -> 0() 1185.67/297.41 , a__half(dbl(X)) -> mark(X) } 1185.67/297.41 Weak DPs: 1185.67/297.41 { a__sqr^#(0()) -> c_5() 1185.67/297.41 , mark^#(0()) -> c_10() 1185.67/297.41 , mark^#(nil()) -> c_11() 1185.67/297.41 , a__first^#(0(), X) -> c_25() 1185.67/297.41 , a__dbl^#(0()) -> c_22() 1185.67/297.41 , a__half^#(s(0())) -> c_28() 1185.67/297.41 , a__half^#(0()) -> c_29() } 1185.67/297.41 Obligation: 1185.67/297.41 runtime complexity 1185.67/297.41 Answer: 1185.67/297.41 MAYBE 1185.67/297.41 1185.67/297.41 Empty strict component of the problem is NOT empty. 1185.67/297.41 1185.67/297.41 1185.67/297.41 Arrrr.. 1186.46/298.13 EOF