We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { a__f(X1, X2) -> f(X1, X2) , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) , mark(g(X)) -> g(mark(X)) , mark(f(X1, X2)) -> a__f(mark(X1), X2) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__f) = {1}, Uargs(g) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__f](x1, x2) = [1] x1 + [1] [g](x1) = [1] x1 + [0] [mark](x1) = [0] [f](x1, x2) = [1] x1 + [0] The order satisfies the following ordering constraints: [a__f(X1, X2)] = [1] X1 + [1] > [1] X1 + [0] = [f(X1, X2)] [a__f(g(X), Y)] = [1] X + [1] >= [1] = [a__f(mark(X), f(g(X), Y))] [mark(g(X))] = [0] >= [0] = [g(mark(X))] [mark(f(X1, X2))] = [0] ? [1] = [a__f(mark(X1), X2)] 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 YES(O(1),O(n^2)). Strict Trs: { a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) , mark(g(X)) -> g(mark(X)) , mark(f(X1, X2)) -> a__f(mark(X1), X2) } Weak Trs: { a__f(X1, X2) -> f(X1, X2) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__f) = {1}, Uargs(g) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__f](x1, x2) = [1] x1 + [4] [g](x1) = [1] x1 + [4] [mark](x1) = [1] [f](x1, x2) = [1] x1 + [0] The order satisfies the following ordering constraints: [a__f(X1, X2)] = [1] X1 + [4] > [1] X1 + [0] = [f(X1, X2)] [a__f(g(X), Y)] = [1] X + [8] > [5] = [a__f(mark(X), f(g(X), Y))] [mark(g(X))] = [1] ? [5] = [g(mark(X))] [mark(f(X1, X2))] = [1] ? [5] = [a__f(mark(X1), X2)] 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 YES(O(1),O(n^2)). Strict Trs: { mark(g(X)) -> g(mark(X)) , mark(f(X1, X2)) -> a__f(mark(X1), X2) } Weak Trs: { a__f(X1, X2) -> f(X1, X2) , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'matrix interpretation of dimension 2' to orient following rules strictly. Trs: { mark(g(X)) -> g(mark(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 usable: Uargs(a__f) = {1}, Uargs(g) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [a__f](x1, x2) = [1 1] x1 + [4] [0 1] [4] [g](x1) = [1 1] x1 + [5] [0 1] [4] [mark](x1) = [1 1] x1 + [2] [0 1] [3] [f](x1, x2) = [1 1] x1 + [3] [0 1] [4] The order satisfies the following ordering constraints: [a__f(X1, X2)] = [1 1] X1 + [4] [0 1] [4] > [1 1] X1 + [3] [0 1] [4] = [f(X1, X2)] [a__f(g(X), Y)] = [1 2] X + [13] [0 1] [8] > [1 2] X + [9] [0 1] [7] = [a__f(mark(X), f(g(X), Y))] [mark(g(X))] = [1 2] X + [11] [0 1] [7] > [1 2] X + [10] [0 1] [7] = [g(mark(X))] [mark(f(X1, X2))] = [1 2] X1 + [9] [0 1] [7] >= [1 2] X1 + [9] [0 1] [7] = [a__f(mark(X1), X2)] We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { mark(f(X1, X2)) -> a__f(mark(X1), X2) } Weak Trs: { a__f(X1, X2) -> f(X1, X2) , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) , mark(g(X)) -> g(mark(X)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'matrix interpretation of dimension 2' to orient following rules strictly. Trs: { mark(f(X1, X2)) -> a__f(mark(X1), X2) } 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 usable: Uargs(a__f) = {1}, Uargs(g) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [a__f](x1, x2) = [1 4] x1 + [0] [0 1] [1] [g](x1) = [1 4] x1 + [0] [0 1] [0] [mark](x1) = [1 4] x1 + [0] [0 1] [0] [f](x1, x2) = [1 4] x1 + [0] [0 1] [1] The order satisfies the following ordering constraints: [a__f(X1, X2)] = [1 4] X1 + [0] [0 1] [1] >= [1 4] X1 + [0] [0 1] [1] = [f(X1, X2)] [a__f(g(X), Y)] = [1 8] X + [0] [0 1] [1] >= [1 8] X + [0] [0 1] [1] = [a__f(mark(X), f(g(X), Y))] [mark(g(X))] = [1 8] X + [0] [0 1] [0] >= [1 8] X + [0] [0 1] [0] = [g(mark(X))] [mark(f(X1, X2))] = [1 8] X1 + [4] [0 1] [1] > [1 8] X1 + [0] [0 1] [1] = [a__f(mark(X1), X2)] We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { a__f(X1, X2) -> f(X1, X2) , a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y)) , mark(g(X)) -> g(mark(X)) , mark(f(X1, X2)) -> a__f(mark(X1), X2) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^2))