MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { terms(N) -> cons(recip(sqr(N)), terms(s(N))) , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , sqr(0()) -> 0() , add(s(X), Y) -> s(add(X, Y)) , add(0(), X) -> X , dbl(s(X)) -> s(s(dbl(X))) , dbl(0()) -> 0() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , first(0(), X) -> nil() } 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: The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(cons) = {1, 2}, Uargs(recip) = {1}, Uargs(s) = {1}, Uargs(add) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [terms](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x1 + [1] x2 + [0] [recip](x1) = [1] x1 + [0] [sqr](x1) = [0] [s](x1) = [1] x1 + [0] [0] = [4] [add](x1, x2) = [1] x1 + [1] x2 + [0] [dbl](x1) = [0] [first](x1, x2) = [1] x2 + [0] [nil] = [7] The following symbols are considered usable {terms, sqr, add, dbl, first} The order satisfies the following ordering constraints: [terms(N)] = [1] N + [0] >= [1] N + [0] = [cons(recip(sqr(N)), terms(s(N)))] [sqr(s(X))] = [0] >= [0] = [s(add(sqr(X), dbl(X)))] [sqr(0())] = [0] ? [4] = [0()] [add(s(X), Y)] = [1] X + [1] Y + [0] >= [1] X + [1] Y + [0] = [s(add(X, Y))] [add(0(), X)] = [1] X + [4] > [1] X + [0] = [X] [dbl(s(X))] = [0] >= [0] = [s(s(dbl(X)))] [dbl(0())] = [0] ? [4] = [0()] [first(s(X), cons(Y, Z))] = [1] Y + [1] Z + [0] >= [1] Y + [1] Z + [0] = [cons(Y, first(X, Z))] [first(0(), X)] = [1] X + [0] ? [7] = [nil()] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { terms(N) -> cons(recip(sqr(N)), terms(s(N))) , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , sqr(0()) -> 0() , add(s(X), Y) -> s(add(X, Y)) , dbl(s(X)) -> s(s(dbl(X))) , dbl(0()) -> 0() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , first(0(), X) -> nil() } Weak Trs: { add(0(), X) -> X } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(cons) = {1, 2}, Uargs(recip) = {1}, Uargs(s) = {1}, Uargs(add) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [terms](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x1 + [1] x2 + [0] [recip](x1) = [1] x1 + [0] [sqr](x1) = [0] [s](x1) = [1] x1 + [0] [0] = [7] [add](x1, x2) = [1] x1 + [1] x2 + [0] [dbl](x1) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [1] [nil] = [7] The following symbols are considered usable {terms, sqr, add, dbl, first} The order satisfies the following ordering constraints: [terms(N)] = [1] N + [0] >= [1] N + [0] = [cons(recip(sqr(N)), terms(s(N)))] [sqr(s(X))] = [0] >= [0] = [s(add(sqr(X), dbl(X)))] [sqr(0())] = [0] ? [7] = [0()] [add(s(X), Y)] = [1] X + [1] Y + [0] >= [1] X + [1] Y + [0] = [s(add(X, Y))] [add(0(), X)] = [1] X + [7] > [1] X + [0] = [X] [dbl(s(X))] = [0] >= [0] = [s(s(dbl(X)))] [dbl(0())] = [0] ? [7] = [0()] [first(s(X), cons(Y, Z))] = [1] X + [1] Y + [1] Z + [1] >= [1] X + [1] Y + [1] Z + [1] = [cons(Y, first(X, Z))] [first(0(), X)] = [1] X + [8] > [7] = [nil()] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { terms(N) -> cons(recip(sqr(N)), terms(s(N))) , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , sqr(0()) -> 0() , add(s(X), Y) -> s(add(X, Y)) , dbl(s(X)) -> s(s(dbl(X))) , dbl(0()) -> 0() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) } Weak Trs: { add(0(), X) -> X , first(0(), X) -> nil() } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(cons) = {1, 2}, Uargs(recip) = {1}, Uargs(s) = {1}, Uargs(add) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [terms](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x1 + [1] x2 + [0] [recip](x1) = [1] x1 + [0] [sqr](x1) = [0] [s](x1) = [1] x1 + [0] [0] = [3] [add](x1, x2) = [1] x1 + [1] x2 + [0] [dbl](x1) = [4] [first](x1, x2) = [1] x1 + [1] x2 + [0] [nil] = [3] The following symbols are considered usable {terms, sqr, add, dbl, first} The order satisfies the following ordering constraints: [terms(N)] = [1] N + [0] >= [1] N + [0] = [cons(recip(sqr(N)), terms(s(N)))] [sqr(s(X))] = [0] ? [4] = [s(add(sqr(X), dbl(X)))] [sqr(0())] = [0] ? [3] = [0()] [add(s(X), Y)] = [1] X + [1] Y + [0] >= [1] X + [1] Y + [0] = [s(add(X, Y))] [add(0(), X)] = [1] X + [3] > [1] X + [0] = [X] [dbl(s(X))] = [4] >= [4] = [s(s(dbl(X)))] [dbl(0())] = [4] > [3] = [0()] [first(s(X), cons(Y, Z))] = [1] X + [1] Y + [1] Z + [0] >= [1] X + [1] Y + [1] Z + [0] = [cons(Y, first(X, Z))] [first(0(), X)] = [1] X + [3] >= [3] = [nil()] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { terms(N) -> cons(recip(sqr(N)), terms(s(N))) , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , sqr(0()) -> 0() , add(s(X), Y) -> s(add(X, Y)) , dbl(s(X)) -> s(s(dbl(X))) , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) } Weak Trs: { add(0(), X) -> X , dbl(0()) -> 0() , first(0(), X) -> nil() } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(cons) = {1, 2}, Uargs(recip) = {1}, Uargs(s) = {1}, Uargs(add) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [terms](x1) = [0] [cons](x1, x2) = [1] x1 + [1] x2 + [0] [recip](x1) = [1] x1 + [0] [sqr](x1) = [0] [s](x1) = [1] x1 + [2] [0] = [2] [add](x1, x2) = [1] x1 + [1] x2 + [0] [dbl](x1) = [2] [first](x1, x2) = [1] x1 + [1] x2 + [2] [nil] = [4] The following symbols are considered usable {terms, sqr, add, dbl, first} The order satisfies the following ordering constraints: [terms(N)] = [0] >= [0] = [cons(recip(sqr(N)), terms(s(N)))] [sqr(s(X))] = [0] ? [4] = [s(add(sqr(X), dbl(X)))] [sqr(0())] = [0] ? [2] = [0()] [add(s(X), Y)] = [1] X + [1] Y + [2] >= [1] X + [1] Y + [2] = [s(add(X, Y))] [add(0(), X)] = [1] X + [2] > [1] X + [0] = [X] [dbl(s(X))] = [2] ? [6] = [s(s(dbl(X)))] [dbl(0())] = [2] >= [2] = [0()] [first(s(X), cons(Y, Z))] = [1] X + [1] Y + [1] Z + [4] > [1] X + [1] Y + [1] Z + [2] = [cons(Y, first(X, Z))] [first(0(), X)] = [1] X + [4] >= [4] = [nil()] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { terms(N) -> cons(recip(sqr(N)), terms(s(N))) , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , sqr(0()) -> 0() , add(s(X), Y) -> s(add(X, Y)) , dbl(s(X)) -> s(s(dbl(X))) } Weak Trs: { add(0(), X) -> X , dbl(0()) -> 0() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , first(0(), X) -> nil() } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(cons) = {1, 2}, Uargs(recip) = {1}, Uargs(s) = {1}, Uargs(add) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [terms](x1) = [1] x1 + [2] [cons](x1, x2) = [1] x1 + [1] x2 + [0] [recip](x1) = [1] x1 + [4] [sqr](x1) = [2] [s](x1) = [1] x1 + [0] [0] = [0] [add](x1, x2) = [1] x1 + [1] x2 + [0] [dbl](x1) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [1] [nil] = [1] The following symbols are considered usable {terms, sqr, add, dbl, first} The order satisfies the following ordering constraints: [terms(N)] = [1] N + [2] ? [1] N + [8] = [cons(recip(sqr(N)), terms(s(N)))] [sqr(s(X))] = [2] >= [2] = [s(add(sqr(X), dbl(X)))] [sqr(0())] = [2] > [0] = [0()] [add(s(X), Y)] = [1] X + [1] Y + [0] >= [1] X + [1] Y + [0] = [s(add(X, Y))] [add(0(), X)] = [1] X + [0] >= [1] X + [0] = [X] [dbl(s(X))] = [0] >= [0] = [s(s(dbl(X)))] [dbl(0())] = [0] >= [0] = [0()] [first(s(X), cons(Y, Z))] = [1] X + [1] Y + [1] Z + [1] >= [1] X + [1] Y + [1] Z + [1] = [cons(Y, first(X, Z))] [first(0(), X)] = [1] X + [1] >= [1] = [nil()] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { terms(N) -> cons(recip(sqr(N)), terms(s(N))) , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(s(X), Y) -> s(add(X, Y)) , dbl(s(X)) -> s(s(dbl(X))) } Weak Trs: { sqr(0()) -> 0() , add(0(), X) -> X , dbl(0()) -> 0() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , first(0(), X) -> nil() } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Fastest' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 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 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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { terms^#(N) -> c_1(sqr^#(N), terms^#(s(N))) , sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X))) , sqr^#(0()) -> c_3() , add^#(s(X), Y) -> c_4(add^#(X, Y)) , add^#(0(), X) -> c_5(X) , dbl^#(s(X)) -> c_6(dbl^#(X)) , dbl^#(0()) -> c_7() , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , first^#(0(), X) -> c_9() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { terms^#(N) -> c_1(sqr^#(N), terms^#(s(N))) , sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X))) , sqr^#(0()) -> c_3() , add^#(s(X), Y) -> c_4(add^#(X, Y)) , add^#(0(), X) -> c_5(X) , dbl^#(s(X)) -> c_6(dbl^#(X)) , dbl^#(0()) -> c_7() , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , first^#(0(), X) -> c_9() } Strict Trs: { terms(N) -> cons(recip(sqr(N)), terms(s(N))) , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , sqr(0()) -> 0() , add(s(X), Y) -> s(add(X, Y)) , add(0(), X) -> X , dbl(s(X)) -> s(s(dbl(X))) , dbl(0()) -> 0() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , first(0(), X) -> nil() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,7,9} by applications of Pre({3,7,9}) = {1,5,6,8}. Here rules are labeled as follows: DPs: { 1: terms^#(N) -> c_1(sqr^#(N), terms^#(s(N))) , 2: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X))) , 3: sqr^#(0()) -> c_3() , 4: add^#(s(X), Y) -> c_4(add^#(X, Y)) , 5: add^#(0(), X) -> c_5(X) , 6: dbl^#(s(X)) -> c_6(dbl^#(X)) , 7: dbl^#(0()) -> c_7() , 8: first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , 9: first^#(0(), X) -> c_9() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { terms^#(N) -> c_1(sqr^#(N), terms^#(s(N))) , sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X))) , add^#(s(X), Y) -> c_4(add^#(X, Y)) , add^#(0(), X) -> c_5(X) , dbl^#(s(X)) -> c_6(dbl^#(X)) , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) } Strict Trs: { terms(N) -> cons(recip(sqr(N)), terms(s(N))) , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , sqr(0()) -> 0() , add(s(X), Y) -> s(add(X, Y)) , add(0(), X) -> X , dbl(s(X)) -> s(s(dbl(X))) , dbl(0()) -> 0() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , first(0(), X) -> nil() } Weak DPs: { sqr^#(0()) -> c_3() , dbl^#(0()) -> c_7() , first^#(0(), X) -> c_9() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..