MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__f(X, g(X), Y) -> a__f(Y, Y, Y) , a__g(X) -> g(X) , a__g(b()) -> c() , a__b() -> b() , a__b() -> c() , mark(g(X)) -> a__g(mark(X)) , mark(b()) -> a__b() , mark(c()) -> c() , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) } 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(g) = {1}, Uargs(a__g) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__f](x1, x2, x3) = [1] x3 + [5] [g](x1) = [1] x1 + [7] [a__g](x1) = [1] x1 + [7] [b] = [7] [c] = [7] [a__b] = [5] [mark](x1) = [1] x1 + [7] [f](x1, x2, x3) = [1] x3 + [7] The following symbols are considered usable {a__f, a__g, a__b, mark} The order satisfies the following ordering constraints: [a__f(X1, X2, X3)] = [1] X3 + [5] ? [1] X3 + [7] = [f(X1, X2, X3)] [a__f(X, g(X), Y)] = [1] Y + [5] >= [1] Y + [5] = [a__f(Y, Y, Y)] [a__g(X)] = [1] X + [7] >= [1] X + [7] = [g(X)] [a__g(b())] = [14] > [7] = [c()] [a__b()] = [5] ? [7] = [b()] [a__b()] = [5] ? [7] = [c()] [mark(g(X))] = [1] X + [14] >= [1] X + [14] = [a__g(mark(X))] [mark(b())] = [14] > [5] = [a__b()] [mark(c())] = [14] > [7] = [c()] [mark(f(X1, X2, X3))] = [1] X3 + [14] > [1] X3 + [5] = [a__f(X1, X2, X3)] 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: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__f(X, g(X), Y) -> a__f(Y, Y, Y) , a__g(X) -> g(X) , a__b() -> b() , a__b() -> c() , mark(g(X)) -> a__g(mark(X)) } Weak Trs: { a__g(b()) -> c() , mark(b()) -> a__b() , mark(c()) -> c() , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(g) = {1}, Uargs(a__g) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__f](x1, x2, x3) = [1] x3 + [7] [g](x1) = [1] x1 + [7] [a__g](x1) = [1] x1 + [7] [b] = [3] [c] = [5] [a__b] = [6] [mark](x1) = [1] x1 + [7] [f](x1, x2, x3) = [1] x3 + [5] The following symbols are considered usable {a__f, a__g, a__b, mark} The order satisfies the following ordering constraints: [a__f(X1, X2, X3)] = [1] X3 + [7] > [1] X3 + [5] = [f(X1, X2, X3)] [a__f(X, g(X), Y)] = [1] Y + [7] >= [1] Y + [7] = [a__f(Y, Y, Y)] [a__g(X)] = [1] X + [7] >= [1] X + [7] = [g(X)] [a__g(b())] = [10] > [5] = [c()] [a__b()] = [6] > [3] = [b()] [a__b()] = [6] > [5] = [c()] [mark(g(X))] = [1] X + [14] >= [1] X + [14] = [a__g(mark(X))] [mark(b())] = [10] > [6] = [a__b()] [mark(c())] = [12] > [5] = [c()] [mark(f(X1, X2, X3))] = [1] X3 + [12] > [1] X3 + [7] = [a__f(X1, X2, X3)] 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: { a__f(X, g(X), Y) -> a__f(Y, Y, Y) , a__g(X) -> g(X) , mark(g(X)) -> a__g(mark(X)) } Weak Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__g(b()) -> c() , a__b() -> b() , a__b() -> c() , mark(b()) -> a__b() , mark(c()) -> c() , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(g) = {1}, Uargs(a__g) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__f](x1, x2, x3) = [1] x3 + [7] [g](x1) = [1] x1 + [0] [a__g](x1) = [1] x1 + [1] [b] = [7] [c] = [4] [a__b] = [7] [mark](x1) = [1] x1 + [0] [f](x1, x2, x3) = [1] x3 + [7] The following symbols are considered usable {a__f, a__g, a__b, mark} The order satisfies the following ordering constraints: [a__f(X1, X2, X3)] = [1] X3 + [7] >= [1] X3 + [7] = [f(X1, X2, X3)] [a__f(X, g(X), Y)] = [1] Y + [7] >= [1] Y + [7] = [a__f(Y, Y, Y)] [a__g(X)] = [1] X + [1] > [1] X + [0] = [g(X)] [a__g(b())] = [8] > [4] = [c()] [a__b()] = [7] >= [7] = [b()] [a__b()] = [7] > [4] = [c()] [mark(g(X))] = [1] X + [0] ? [1] X + [1] = [a__g(mark(X))] [mark(b())] = [7] >= [7] = [a__b()] [mark(c())] = [4] >= [4] = [c()] [mark(f(X1, X2, X3))] = [1] X3 + [7] >= [1] X3 + [7] = [a__f(X1, X2, X3)] 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: { a__f(X, g(X), Y) -> a__f(Y, Y, Y) , mark(g(X)) -> a__g(mark(X)) } Weak Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__g(X) -> g(X) , a__g(b()) -> c() , a__b() -> b() , a__b() -> c() , mark(b()) -> a__b() , mark(c()) -> c() , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) } Obligation: runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { mark(g(X)) -> a__g(mark(X)) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(g) = {1}, Uargs(a__g) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [a__f](x1, x2, x3) = [0] [g](x1) = [1] x1 + [4] [a__g](x1) = [1] x1 + [4] [b] = [4] [c] = [0] [a__b] = [4] [mark](x1) = [2] x1 + [0] [f](x1, x2, x3) = [0] The following symbols are considered usable {a__f, a__g, a__b, mark} The order satisfies the following ordering constraints: [a__f(X1, X2, X3)] = [0] >= [0] = [f(X1, X2, X3)] [a__f(X, g(X), Y)] = [0] >= [0] = [a__f(Y, Y, Y)] [a__g(X)] = [1] X + [4] >= [1] X + [4] = [g(X)] [a__g(b())] = [8] > [0] = [c()] [a__b()] = [4] >= [4] = [b()] [a__b()] = [4] > [0] = [c()] [mark(g(X))] = [2] X + [8] > [2] X + [4] = [a__g(mark(X))] [mark(b())] = [8] > [4] = [a__b()] [mark(c())] = [0] >= [0] = [c()] [mark(f(X1, X2, X3))] = [0] >= [0] = [a__f(X1, X2, X3)] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { a__f(X, g(X), Y) -> a__f(Y, Y, Y) } Weak Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__g(X) -> g(X) , a__g(b()) -> c() , a__b() -> b() , a__b() -> c() , mark(g(X)) -> a__g(mark(X)) , mark(b()) -> a__b() , mark(c()) -> c() , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) } 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 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: { a__f^#(X1, X2, X3) -> c_1(X1, X2, X3) , a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y)) , a__g^#(X) -> c_3(X) , a__g^#(b()) -> c_4() , a__b^#() -> c_5() , a__b^#() -> c_6() , mark^#(g(X)) -> c_7(a__g^#(mark(X))) , mark^#(b()) -> c_8(a__b^#()) , mark^#(c()) -> c_9() , mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__f^#(X1, X2, X3) -> c_1(X1, X2, X3) , a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y)) , a__g^#(X) -> c_3(X) , a__g^#(b()) -> c_4() , a__b^#() -> c_5() , a__b^#() -> c_6() , mark^#(g(X)) -> c_7(a__g^#(mark(X))) , mark^#(b()) -> c_8(a__b^#()) , mark^#(c()) -> c_9() , mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3)) } Strict Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__f(X, g(X), Y) -> a__f(Y, Y, Y) , a__g(X) -> g(X) , a__g(b()) -> c() , a__b() -> b() , a__b() -> c() , mark(g(X)) -> a__g(mark(X)) , mark(b()) -> a__b() , mark(c()) -> c() , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {4,5,6,9} by applications of Pre({4,5,6,9}) = {1,3,7,8}. Here rules are labeled as follows: DPs: { 1: a__f^#(X1, X2, X3) -> c_1(X1, X2, X3) , 2: a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y)) , 3: a__g^#(X) -> c_3(X) , 4: a__g^#(b()) -> c_4() , 5: a__b^#() -> c_5() , 6: a__b^#() -> c_6() , 7: mark^#(g(X)) -> c_7(a__g^#(mark(X))) , 8: mark^#(b()) -> c_8(a__b^#()) , 9: mark^#(c()) -> c_9() , 10: mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__f^#(X1, X2, X3) -> c_1(X1, X2, X3) , a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y)) , a__g^#(X) -> c_3(X) , mark^#(g(X)) -> c_7(a__g^#(mark(X))) , mark^#(b()) -> c_8(a__b^#()) , mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3)) } Strict Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__f(X, g(X), Y) -> a__f(Y, Y, Y) , a__g(X) -> g(X) , a__g(b()) -> c() , a__b() -> b() , a__b() -> c() , mark(g(X)) -> a__g(mark(X)) , mark(b()) -> a__b() , mark(c()) -> c() , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) } Weak DPs: { a__g^#(b()) -> c_4() , a__b^#() -> c_5() , a__b^#() -> c_6() , mark^#(c()) -> c_9() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {5} by applications of Pre({5}) = {1,3}. Here rules are labeled as follows: DPs: { 1: a__f^#(X1, X2, X3) -> c_1(X1, X2, X3) , 2: a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y)) , 3: a__g^#(X) -> c_3(X) , 4: mark^#(g(X)) -> c_7(a__g^#(mark(X))) , 5: mark^#(b()) -> c_8(a__b^#()) , 6: mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3)) , 7: a__g^#(b()) -> c_4() , 8: a__b^#() -> c_5() , 9: a__b^#() -> c_6() , 10: mark^#(c()) -> c_9() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { a__f^#(X1, X2, X3) -> c_1(X1, X2, X3) , a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y)) , a__g^#(X) -> c_3(X) , mark^#(g(X)) -> c_7(a__g^#(mark(X))) , mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3)) } Strict Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__f(X, g(X), Y) -> a__f(Y, Y, Y) , a__g(X) -> g(X) , a__g(b()) -> c() , a__b() -> b() , a__b() -> c() , mark(g(X)) -> a__g(mark(X)) , mark(b()) -> a__b() , mark(c()) -> c() , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) } Weak DPs: { a__g^#(b()) -> c_4() , a__b^#() -> c_5() , a__b^#() -> c_6() , mark^#(b()) -> c_8(a__b^#()) , mark^#(c()) -> c_9() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..