MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { null(nil()) -> true() , null(add(n, x)) -> false() , tail(nil()) -> nil() , tail(add(n, x)) -> x , head(add(n, x)) -> n , app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , reverse(nil()) -> nil() , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) , shuffle(x) -> shuff(x, nil()) , shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil()))) , if(true(), x, y, z) -> y , if(false(), x, y, z) -> shuff(reverse(tail(x)), 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: { null^#(nil()) -> c_1() , null^#(add(n, x)) -> c_2() , tail^#(nil()) -> c_3() , tail^#(add(n, x)) -> c_4(x) , head^#(add(n, x)) -> c_5(n) , app^#(nil(), y) -> c_6(y) , app^#(add(n, x), y) -> c_7(n, app^#(x, y)) , reverse^#(nil()) -> c_8() , reverse^#(add(n, x)) -> c_9(app^#(reverse(x), add(n, nil()))) , shuffle^#(x) -> c_10(shuff^#(x, nil())) , shuff^#(x, y) -> c_11(if^#(null(x), x, y, app(y, add(head(x), nil())))) , if^#(true(), x, y, z) -> c_12(y) , if^#(false(), x, y, z) -> c_13(shuff^#(reverse(tail(x)), z)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { null^#(nil()) -> c_1() , null^#(add(n, x)) -> c_2() , tail^#(nil()) -> c_3() , tail^#(add(n, x)) -> c_4(x) , head^#(add(n, x)) -> c_5(n) , app^#(nil(), y) -> c_6(y) , app^#(add(n, x), y) -> c_7(n, app^#(x, y)) , reverse^#(nil()) -> c_8() , reverse^#(add(n, x)) -> c_9(app^#(reverse(x), add(n, nil()))) , shuffle^#(x) -> c_10(shuff^#(x, nil())) , shuff^#(x, y) -> c_11(if^#(null(x), x, y, app(y, add(head(x), nil())))) , if^#(true(), x, y, z) -> c_12(y) , if^#(false(), x, y, z) -> c_13(shuff^#(reverse(tail(x)), z)) } Strict Trs: { null(nil()) -> true() , null(add(n, x)) -> false() , tail(nil()) -> nil() , tail(add(n, x)) -> x , head(add(n, x)) -> n , app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , reverse(nil()) -> nil() , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) , shuffle(x) -> shuff(x, nil()) , shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil()))) , if(true(), x, y, z) -> y , if(false(), x, y, z) -> shuff(reverse(tail(x)), z) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,3,8} by applications of Pre({1,2,3,8}) = {4,5,6,7,12}. Here rules are labeled as follows: DPs: { 1: null^#(nil()) -> c_1() , 2: null^#(add(n, x)) -> c_2() , 3: tail^#(nil()) -> c_3() , 4: tail^#(add(n, x)) -> c_4(x) , 5: head^#(add(n, x)) -> c_5(n) , 6: app^#(nil(), y) -> c_6(y) , 7: app^#(add(n, x), y) -> c_7(n, app^#(x, y)) , 8: reverse^#(nil()) -> c_8() , 9: reverse^#(add(n, x)) -> c_9(app^#(reverse(x), add(n, nil()))) , 10: shuffle^#(x) -> c_10(shuff^#(x, nil())) , 11: shuff^#(x, y) -> c_11(if^#(null(x), x, y, app(y, add(head(x), nil())))) , 12: if^#(true(), x, y, z) -> c_12(y) , 13: if^#(false(), x, y, z) -> c_13(shuff^#(reverse(tail(x)), z)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { tail^#(add(n, x)) -> c_4(x) , head^#(add(n, x)) -> c_5(n) , app^#(nil(), y) -> c_6(y) , app^#(add(n, x), y) -> c_7(n, app^#(x, y)) , reverse^#(add(n, x)) -> c_9(app^#(reverse(x), add(n, nil()))) , shuffle^#(x) -> c_10(shuff^#(x, nil())) , shuff^#(x, y) -> c_11(if^#(null(x), x, y, app(y, add(head(x), nil())))) , if^#(true(), x, y, z) -> c_12(y) , if^#(false(), x, y, z) -> c_13(shuff^#(reverse(tail(x)), z)) } Strict Trs: { null(nil()) -> true() , null(add(n, x)) -> false() , tail(nil()) -> nil() , tail(add(n, x)) -> x , head(add(n, x)) -> n , app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , reverse(nil()) -> nil() , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) , shuffle(x) -> shuff(x, nil()) , shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil()))) , if(true(), x, y, z) -> y , if(false(), x, y, z) -> shuff(reverse(tail(x)), z) } Weak DPs: { null^#(nil()) -> c_1() , null^#(add(n, x)) -> c_2() , tail^#(nil()) -> c_3() , reverse^#(nil()) -> c_8() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..