MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { sqr(0()) -> 0() , sqr(s(x)) -> s(+(sqr(x), double(x))) , sqr(s(x)) -> +(sqr(x), s(double(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , double(0()) -> 0() , double(s(x)) -> s(s(double(x))) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) '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(s) = {1}, Uargs(+) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [sqr](x1) = [0] [0] = [1] [s](x1) = [1] x1 + [0] [+](x1, x2) = [1] x1 + [1] x2 + [0] [double](x1) = [0] The following symbols are considered usable {sqr, +, double} The order satisfies the following ordering constraints: [sqr(0())] = [0] ? [1] = [0()] [sqr(s(x))] = [0] >= [0] = [s(+(sqr(x), double(x)))] [sqr(s(x))] = [0] >= [0] = [+(sqr(x), s(double(x)))] [+(x, 0())] = [1] x + [1] > [1] x + [0] = [x] [+(x, s(y))] = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = [s(+(x, y))] [double(0())] = [0] ? [1] = [0()] [double(s(x))] = [0] >= [0] = [s(s(double(x)))] 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(+(sqr(x), double(x))) , sqr(s(x)) -> +(sqr(x), s(double(x))) , +(x, s(y)) -> s(+(x, y)) , double(0()) -> 0() , double(s(x)) -> s(s(double(x))) } Weak Trs: { +(x, 0()) -> x } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(+) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [sqr](x1) = [0] [0] = [0] [s](x1) = [1] x1 + [0] [+](x1, x2) = [1] x1 + [1] x2 + [0] [double](x1) = [1] The following symbols are considered usable {sqr, +, double} The order satisfies the following ordering constraints: [sqr(0())] = [0] >= [0] = [0()] [sqr(s(x))] = [0] ? [1] = [s(+(sqr(x), double(x)))] [sqr(s(x))] = [0] ? [1] = [+(sqr(x), s(double(x)))] [+(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [+(x, s(y))] = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = [s(+(x, y))] [double(0())] = [1] > [0] = [0()] [double(s(x))] = [1] >= [1] = [s(s(double(x)))] 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(+(sqr(x), double(x))) , sqr(s(x)) -> +(sqr(x), s(double(x))) , +(x, s(y)) -> s(+(x, y)) , double(s(x)) -> s(s(double(x))) } Weak Trs: { +(x, 0()) -> x , double(0()) -> 0() } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(+) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [sqr](x1) = [1] x1 + [4] [0] = [4] [s](x1) = [1] x1 + [0] [+](x1, x2) = [1] x1 + [1] x2 + [0] [double](x1) = [4] The following symbols are considered usable {sqr, +, double} The order satisfies the following ordering constraints: [sqr(0())] = [8] > [4] = [0()] [sqr(s(x))] = [1] x + [4] ? [1] x + [8] = [s(+(sqr(x), double(x)))] [sqr(s(x))] = [1] x + [4] ? [1] x + [8] = [+(sqr(x), s(double(x)))] [+(x, 0())] = [1] x + [4] > [1] x + [0] = [x] [+(x, s(y))] = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = [s(+(x, y))] [double(0())] = [4] >= [4] = [0()] [double(s(x))] = [4] >= [4] = [s(s(double(x)))] 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(+(sqr(x), double(x))) , sqr(s(x)) -> +(sqr(x), s(double(x))) , +(x, s(y)) -> s(+(x, y)) , double(s(x)) -> s(s(double(x))) } Weak Trs: { sqr(0()) -> 0() , +(x, 0()) -> x , double(0()) -> 0() } 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(s) = {1}, Uargs(+) = {1, 2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [sqr](x1) = [0 1] x1 + [0] [0 1] [0] [0] = [0] [0] [s](x1) = [1 0] x1 + [0] [0 1] [4] [+](x1, x2) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 0] [0] [double](x1) = [0] [4] The following symbols are considered usable {sqr, +, double} The order satisfies the following ordering constraints: [sqr(0())] = [0] [0] >= [0] [0] = [0()] [sqr(s(x))] = [0 1] x + [4] [0 1] [4] > [0 1] x + [0] [0 1] [4] = [s(+(sqr(x), double(x)))] [sqr(s(x))] = [0 1] x + [4] [0 1] [4] > [0 1] x + [0] [0 1] [0] = [+(sqr(x), s(double(x)))] [+(x, 0())] = [1 0] x + [0] [0 1] [0] >= [1 0] x + [0] [0 1] [0] = [x] [+(x, s(y))] = [1 0] x + [1 0] y + [0] [0 1] [0 0] [0] ? [1 0] x + [1 0] y + [0] [0 1] [0 0] [4] = [s(+(x, y))] [double(0())] = [0] [4] >= [0] [0] = [0()] [double(s(x))] = [0] [4] ? [0] [12] = [s(s(double(x)))] 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: { +(x, s(y)) -> s(+(x, y)) , double(s(x)) -> s(s(double(x))) } Weak Trs: { sqr(0()) -> 0() , sqr(s(x)) -> s(+(sqr(x), double(x))) , sqr(s(x)) -> +(sqr(x), s(double(x))) , +(x, 0()) -> x , double(0()) -> 0() } 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(+(sqr(x), double(x))) , +(x, s(y)) -> s(+(x, y)) } 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(s) = {1}, Uargs(+) = {1, 2} TcT has computed the following constructor-restricted polynomial interpretation. [sqr](x1) = 2*x1^2 [0]() = 0 [s](x1) = 1 + x1 [+](x1, x2) = x1 + 2*x2 [double](x1) = 2*x1 The following symbols are considered usable {sqr, +, double} This order satisfies the following ordering constraints. [sqr(0())] = >= = [0()] [sqr(s(x))] = 2 + 4*x + 2*x^2 > 1 + 2*x^2 + 4*x = [s(+(sqr(x), double(x)))] [sqr(s(x))] = 2 + 4*x + 2*x^2 >= 2*x^2 + 2 + 4*x = [+(sqr(x), s(double(x)))] [+(x, 0())] = x >= x = [x] [+(x, s(y))] = x + 2 + 2*y > 1 + x + 2*y = [s(+(x, y))] [double(0())] = >= = [0()] [double(s(x))] = 2 + 2*x >= 2 + 2*x = [s(s(double(x)))] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { sqr(s(x)) -> +(sqr(x), s(double(x))) , double(s(x)) -> s(s(double(x))) } Weak Trs: { sqr(0()) -> 0() , sqr(s(x)) -> s(+(sqr(x), double(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , double(0()) -> 0() } Obligation: runtime complexity Answer: MAYBE We use the processor 'polynomial interpretation' to orient following rules strictly. Trs: { sqr(s(x)) -> +(sqr(x), s(double(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(s) = {1}, Uargs(+) = {1, 2} TcT has computed the following constructor-restricted polynomial interpretation. [sqr](x1) = 2*x1 + x1^2 [0]() = 0 [s](x1) = 1 + x1 [+](x1, x2) = x1 + x2 [double](x1) = 2*x1 The following symbols are considered usable {sqr, +, double} This order satisfies the following ordering constraints. [sqr(0())] = >= = [0()] [sqr(s(x))] = 3 + 4*x + x^2 > 1 + 4*x + x^2 = [s(+(sqr(x), double(x)))] [sqr(s(x))] = 3 + 4*x + x^2 > 4*x + x^2 + 1 = [+(sqr(x), s(double(x)))] [+(x, 0())] = x >= x = [x] [+(x, s(y))] = x + 1 + y >= 1 + x + y = [s(+(x, y))] [double(0())] = >= = [0()] [double(s(x))] = 2 + 2*x >= 2 + 2*x = [s(s(double(x)))] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { double(s(x)) -> s(s(double(x))) } Weak Trs: { sqr(0()) -> 0() , sqr(s(x)) -> s(+(sqr(x), double(x))) , sqr(s(x)) -> +(sqr(x), s(double(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , double(0()) -> 0() } 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) '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) '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 2) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { sqr^#(0()) -> c_1() , sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x))) , sqr^#(s(x)) -> c_3(+^#(sqr(x), s(double(x)))) , +^#(x, 0()) -> c_4(x) , +^#(x, s(y)) -> c_5(+^#(x, y)) , double^#(0()) -> c_6() , double^#(s(x)) -> c_7(double^#(x)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { sqr^#(0()) -> c_1() , sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x))) , sqr^#(s(x)) -> c_3(+^#(sqr(x), s(double(x)))) , +^#(x, 0()) -> c_4(x) , +^#(x, s(y)) -> c_5(+^#(x, y)) , double^#(0()) -> c_6() , double^#(s(x)) -> c_7(double^#(x)) } Strict Trs: { sqr(0()) -> 0() , sqr(s(x)) -> s(+(sqr(x), double(x))) , sqr(s(x)) -> +(sqr(x), s(double(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , double(0()) -> 0() , double(s(x)) -> s(s(double(x))) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,6} by applications of Pre({1,6}) = {4,7}. Here rules are labeled as follows: DPs: { 1: sqr^#(0()) -> c_1() , 2: sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x))) , 3: sqr^#(s(x)) -> c_3(+^#(sqr(x), s(double(x)))) , 4: +^#(x, 0()) -> c_4(x) , 5: +^#(x, s(y)) -> c_5(+^#(x, y)) , 6: double^#(0()) -> c_6() , 7: double^#(s(x)) -> c_7(double^#(x)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x))) , sqr^#(s(x)) -> c_3(+^#(sqr(x), s(double(x)))) , +^#(x, 0()) -> c_4(x) , +^#(x, s(y)) -> c_5(+^#(x, y)) , double^#(s(x)) -> c_7(double^#(x)) } Strict Trs: { sqr(0()) -> 0() , sqr(s(x)) -> s(+(sqr(x), double(x))) , sqr(s(x)) -> +(sqr(x), s(double(x))) , +(x, 0()) -> x , +(x, s(y)) -> s(+(x, y)) , double(0()) -> 0() , double(s(x)) -> s(s(double(x))) } Weak DPs: { sqr^#(0()) -> c_1() , double^#(0()) -> c_6() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..