MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { terms(N) -> cons(recip(sqr(N))) , sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } 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}, 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 + [7] [cons](x1) = [1] x1 + [0] [recip](x1) = [1] x1 + [1] [sqr](x1) = [0] [0] = [0] [s](x1) = [1] x1 + [0] [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 + [7] > [1] = [cons(recip(sqr(N)))] [sqr(0())] = [0] >= [0] = [0()] [sqr(s(X))] = [0] >= [0] = [s(add(sqr(X), dbl(X)))] [add(0(), X)] = [1] X + [0] >= [1] X + [0] = [X] [add(s(X), Y)] = [1] X + [1] Y + [0] >= [1] X + [1] Y + [0] = [s(add(X, Y))] [dbl(0())] = [0] >= [0] = [0()] [dbl(s(X))] = [0] >= [0] = [s(s(dbl(X)))] [first(0(), X)] = [1] X + [0] ? [7] = [nil()] [first(s(X), cons(Y))] = [1] Y + [0] >= [1] Y + [0] = [cons(Y)] 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: { sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } Weak Trs: { terms(N) -> cons(recip(sqr(N))) } 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}, 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 + [7] [cons](x1) = [1] x1 + [0] [recip](x1) = [1] x1 + [5] [sqr](x1) = [0] [0] = [1] [s](x1) = [1] x1 + [0] [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 + [7] > [5] = [cons(recip(sqr(N)))] [sqr(0())] = [0] ? [1] = [0()] [sqr(s(X))] = [0] >= [0] = [s(add(sqr(X), dbl(X)))] [add(0(), X)] = [1] X + [1] > [1] X + [0] = [X] [add(s(X), Y)] = [1] X + [1] Y + [0] >= [1] X + [1] Y + [0] = [s(add(X, Y))] [dbl(0())] = [0] ? [1] = [0()] [dbl(s(X))] = [0] >= [0] = [s(s(dbl(X)))] [first(0(), X)] = [1] X + [0] ? [7] = [nil()] [first(s(X), cons(Y))] = [1] Y + [0] >= [1] Y + [0] = [cons(Y)] 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: { sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } Weak Trs: { terms(N) -> cons(recip(sqr(N))) , 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}, 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 + [7] [cons](x1) = [1] x1 + [0] [recip](x1) = [1] x1 + [7] [sqr](x1) = [0] [0] = [1] [s](x1) = [1] x1 + [0] [add](x1, x2) = [1] x1 + [1] x2 + [0] [dbl](x1) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [0] [nil] = [0] The following symbols are considered usable {terms, sqr, add, dbl, first} The order satisfies the following ordering constraints: [terms(N)] = [1] N + [7] >= [7] = [cons(recip(sqr(N)))] [sqr(0())] = [0] ? [1] = [0()] [sqr(s(X))] = [0] >= [0] = [s(add(sqr(X), dbl(X)))] [add(0(), X)] = [1] X + [1] > [1] X + [0] = [X] [add(s(X), Y)] = [1] X + [1] Y + [0] >= [1] X + [1] Y + [0] = [s(add(X, Y))] [dbl(0())] = [0] ? [1] = [0()] [dbl(s(X))] = [0] >= [0] = [s(s(dbl(X)))] [first(0(), X)] = [1] X + [1] > [0] = [nil()] [first(s(X), cons(Y))] = [1] X + [1] Y + [0] >= [1] Y + [0] = [cons(Y)] 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: { sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , first(s(X), cons(Y)) -> cons(Y) } Weak Trs: { terms(N) -> cons(recip(sqr(N))) , 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}, 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 + [7] [cons](x1) = [1] x1 + [3] [recip](x1) = [1] x1 + [3] [sqr](x1) = [1] x1 + [0] [0] = [6] [s](x1) = [1] x1 + [1] [add](x1, x2) = [1] x1 + [1] x2 + [7] [dbl](x1) = [7] [first](x1, x2) = [1] x1 + [1] x2 + [3] [nil] = [1] The following symbols are considered usable {terms, sqr, add, dbl, first} The order satisfies the following ordering constraints: [terms(N)] = [1] N + [7] > [1] N + [6] = [cons(recip(sqr(N)))] [sqr(0())] = [6] >= [6] = [0()] [sqr(s(X))] = [1] X + [1] ? [1] X + [15] = [s(add(sqr(X), dbl(X)))] [add(0(), X)] = [1] X + [13] > [1] X + [0] = [X] [add(s(X), Y)] = [1] X + [1] Y + [8] >= [1] X + [1] Y + [8] = [s(add(X, Y))] [dbl(0())] = [7] > [6] = [0()] [dbl(s(X))] = [7] ? [9] = [s(s(dbl(X)))] [first(0(), X)] = [1] X + [9] > [1] = [nil()] [first(s(X), cons(Y))] = [1] X + [1] Y + [7] > [1] Y + [3] = [cons(Y)] 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: { sqr(0()) -> 0() , 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: { terms(N) -> cons(recip(sqr(N))) , add(0(), X) -> X , dbl(0()) -> 0() , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } 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}, 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 + [7] [cons](x1) = [1] x1 + [3] [recip](x1) = [1] x1 + [3] [sqr](x1) = [1] x1 + [1] [0] = [0] [s](x1) = [1] x1 + [0] [add](x1, x2) = [1] x1 + [1] x2 + [0] [dbl](x1) = [0] [first](x1, x2) = [1] x1 + [1] x2 + [7] [nil] = [7] The following symbols are considered usable {terms, sqr, add, dbl, first} The order satisfies the following ordering constraints: [terms(N)] = [1] N + [7] >= [1] N + [7] = [cons(recip(sqr(N)))] [sqr(0())] = [1] > [0] = [0()] [sqr(s(X))] = [1] X + [1] >= [1] X + [1] = [s(add(sqr(X), dbl(X)))] [add(0(), X)] = [1] X + [0] >= [1] X + [0] = [X] [add(s(X), Y)] = [1] X + [1] Y + [0] >= [1] X + [1] Y + [0] = [s(add(X, Y))] [dbl(0())] = [0] >= [0] = [0()] [dbl(s(X))] = [0] >= [0] = [s(s(dbl(X)))] [first(0(), X)] = [1] X + [7] >= [7] = [nil()] [first(s(X), cons(Y))] = [1] X + [1] Y + [10] > [1] Y + [3] = [cons(Y)] 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: { 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: { terms(N) -> cons(recip(sqr(N))) , sqr(0()) -> 0() , add(0(), X) -> X , dbl(0()) -> 0() , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } 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: The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(cons) = {1}, 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 1] x1 + [7] [0 0] [7] [cons](x1) = [1 0] x1 + [0] [0 0] [0] [recip](x1) = [1 0] x1 + [4] [0 0] [4] [sqr](x1) = [0 1] x1 + [0] [0 0] [4] [0] = [0] [0] [s](x1) = [1 0] x1 + [0] [0 1] [1] [add](x1, x2) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 1] [0] [dbl](x1) = [0] [2] [first](x1, x2) = [1 0] x2 + [4] [0 0] [0] [nil] = [4] [0] The following symbols are considered usable {terms, sqr, add, dbl, first} The order satisfies the following ordering constraints: [terms(N)] = [0 1] N + [7] [0 0] [7] > [0 1] N + [4] [0 0] [0] = [cons(recip(sqr(N)))] [sqr(0())] = [0] [4] >= [0] [0] = [0()] [sqr(s(X))] = [0 1] X + [1] [0 0] [4] > [0 1] X + [0] [0 0] [3] = [s(add(sqr(X), dbl(X)))] [add(0(), X)] = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = [X] [add(s(X), Y)] = [1 0] X + [1 0] Y + [0] [0 0] [0 1] [0] ? [1 0] X + [1 0] Y + [0] [0 0] [0 1] [1] = [s(add(X, Y))] [dbl(0())] = [0] [2] >= [0] [0] = [0()] [dbl(s(X))] = [0] [2] ? [0] [4] = [s(s(dbl(X)))] [first(0(), X)] = [1 0] X + [4] [0 0] [0] >= [4] [0] = [nil()] [first(s(X), cons(Y))] = [1 0] Y + [4] [0 0] [0] > [1 0] Y + [0] [0 0] [0] = [cons(Y)] 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: { add(s(X), Y) -> s(add(X, Y)) , dbl(s(X)) -> s(s(dbl(X))) } Weak Trs: { terms(N) -> cons(recip(sqr(N))) , sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , dbl(0()) -> 0() , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } 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) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: We use the processor 'polynomial interpretation' to orient following rules strictly. Trs: { sqr(s(X)) -> s(add(sqr(X), dbl(X))) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^2)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are considered usable: Uargs(cons) = {1}, Uargs(recip) = {1}, Uargs(s) = {1}, Uargs(add) = {1, 2} TcT has computed the following constructor-restricted polynomial interpretation. [terms](x1) = 3 + 3*x1 + 3*x1^2 [cons](x1) = x1 [recip](x1) = 3 + x1 [sqr](x1) = x1^2 [0]() = 0 [s](x1) = 2 + x1 [add](x1, x2) = x1 + 2*x2 [dbl](x1) = 2*x1 [first](x1, x2) = 1 + x2 [nil]() = 1 The following symbols are considered usable {terms, sqr, add, dbl, first} This order satisfies the following ordering constraints. [terms(N)] = 3 + 3*N + 3*N^2 >= 3 + N^2 = [cons(recip(sqr(N)))] [sqr(0())] = >= = [0()] [sqr(s(X))] = 4 + 4*X + X^2 > 2 + X^2 + 4*X = [s(add(sqr(X), dbl(X)))] [add(0(), X)] = 2*X >= X = [X] [add(s(X), Y)] = 2 + X + 2*Y >= 2 + X + 2*Y = [s(add(X, Y))] [dbl(0())] = >= = [0()] [dbl(s(X))] = 4 + 2*X >= 4 + 2*X = [s(s(dbl(X)))] [first(0(), X)] = 1 + X >= 1 = [nil()] [first(s(X), cons(Y))] = 1 + Y > Y = [cons(Y)] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { add(s(X), Y) -> s(add(X, Y)) , dbl(s(X)) -> s(s(dbl(X))) } Weak Trs: { terms(N) -> cons(recip(sqr(N))) , sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , dbl(0()) -> 0() , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } 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: 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 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: { terms^#(N) -> c_1(sqr^#(N)) , sqr^#(0()) -> c_2() , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X))) , add^#(0(), X) -> c_4(X) , add^#(s(X), Y) -> c_5(add^#(X, Y)) , dbl^#(0()) -> c_6() , dbl^#(s(X)) -> c_7(dbl^#(X)) , first^#(0(), X) -> c_8() , first^#(s(X), cons(Y)) -> c_9(Y) } 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)) , sqr^#(0()) -> c_2() , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X))) , add^#(0(), X) -> c_4(X) , add^#(s(X), Y) -> c_5(add^#(X, Y)) , dbl^#(0()) -> c_6() , dbl^#(s(X)) -> c_7(dbl^#(X)) , first^#(0(), X) -> c_8() , first^#(s(X), cons(Y)) -> c_9(Y) } Strict Trs: { terms(N) -> cons(recip(sqr(N))) , sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,6,8} by applications of Pre({2,6,8}) = {1,4,7,9}. Here rules are labeled as follows: DPs: { 1: terms^#(N) -> c_1(sqr^#(N)) , 2: sqr^#(0()) -> c_2() , 3: sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X))) , 4: add^#(0(), X) -> c_4(X) , 5: add^#(s(X), Y) -> c_5(add^#(X, Y)) , 6: dbl^#(0()) -> c_6() , 7: dbl^#(s(X)) -> c_7(dbl^#(X)) , 8: first^#(0(), X) -> c_8() , 9: first^#(s(X), cons(Y)) -> c_9(Y) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { terms^#(N) -> c_1(sqr^#(N)) , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X))) , add^#(0(), X) -> c_4(X) , add^#(s(X), Y) -> c_5(add^#(X, Y)) , dbl^#(s(X)) -> c_7(dbl^#(X)) , first^#(s(X), cons(Y)) -> c_9(Y) } Strict Trs: { terms(N) -> cons(recip(sqr(N))) , sqr(0()) -> 0() , sqr(s(X)) -> s(add(sqr(X), dbl(X))) , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , dbl(0()) -> 0() , dbl(s(X)) -> s(s(dbl(X))) , first(0(), X) -> nil() , first(s(X), cons(Y)) -> cons(Y) } Weak DPs: { sqr^#(0()) -> c_2() , dbl^#(0()) -> c_6() , first^#(0(), X) -> c_8() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..