YES(O(1),O(n^1)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__f(a(), X, X) -> a__f(X, a__b(), b()) , a__b() -> a() , a__b() -> b() , mark(a()) -> a() , mark(b()) -> a__b() , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__f) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__f](x1, x2, x3) = [1] x2 + [4] [a] = [0] [a__b] = [0] [b] = [5] [mark](x1) = [0] [f](x1, x2, x3) = [1] x2 + [3] The following symbols are considered usable {a__f, a__b, mark} The order satisfies the following ordering constraints: [a__f(X1, X2, X3)] = [1] X2 + [4] > [1] X2 + [3] = [f(X1, X2, X3)] [a__f(a(), X, X)] = [1] X + [4] >= [4] = [a__f(X, a__b(), b())] [a__b()] = [0] >= [0] = [a()] [a__b()] = [0] ? [5] = [b()] [mark(a())] = [0] >= [0] = [a()] [mark(b())] = [0] >= [0] = [a__b()] [mark(f(X1, X2, X3))] = [0] ? [4] = [a__f(X1, mark(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 YES(O(1),O(n^1)). Strict Trs: { a__f(a(), X, X) -> a__f(X, a__b(), b()) , a__b() -> a() , a__b() -> b() , mark(a()) -> a() , mark(b()) -> a__b() , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) } Weak Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__f) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__f](x1, x2, x3) = [1] x2 + [4] [a] = [0] [a__b] = [4] [b] = [5] [mark](x1) = [0] [f](x1, x2, x3) = [1] x2 + [3] The following symbols are considered usable {a__f, a__b, mark} The order satisfies the following ordering constraints: [a__f(X1, X2, X3)] = [1] X2 + [4] > [1] X2 + [3] = [f(X1, X2, X3)] [a__f(a(), X, X)] = [1] X + [4] ? [8] = [a__f(X, a__b(), b())] [a__b()] = [4] > [0] = [a()] [a__b()] = [4] ? [5] = [b()] [mark(a())] = [0] >= [0] = [a()] [mark(b())] = [0] ? [4] = [a__b()] [mark(f(X1, X2, X3))] = [0] ? [4] = [a__f(X1, mark(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 YES(O(1),O(n^1)). Strict Trs: { a__f(a(), X, X) -> a__f(X, a__b(), b()) , a__b() -> b() , mark(a()) -> a() , mark(b()) -> a__b() , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) } Weak Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__b() -> a() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__f) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__f](x1, x2, x3) = [1] x2 + [0] [a] = [0] [a__b] = [4] [b] = [1] [mark](x1) = [0] [f](x1, x2, x3) = [1] x2 + [0] The following symbols are considered usable {a__f, a__b, mark} The order satisfies the following ordering constraints: [a__f(X1, X2, X3)] = [1] X2 + [0] >= [1] X2 + [0] = [f(X1, X2, X3)] [a__f(a(), X, X)] = [1] X + [0] ? [4] = [a__f(X, a__b(), b())] [a__b()] = [4] > [0] = [a()] [a__b()] = [4] > [1] = [b()] [mark(a())] = [0] >= [0] = [a()] [mark(b())] = [0] ? [4] = [a__b()] [mark(f(X1, X2, X3))] = [0] >= [0] = [a__f(X1, mark(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 YES(O(1),O(n^1)). Strict Trs: { a__f(a(), X, X) -> a__f(X, a__b(), b()) , mark(a()) -> a() , mark(b()) -> a__b() , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) } Weak Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__b() -> a() , a__b() -> b() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__f) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__f](x1, x2, x3) = [1] x2 + [4] [a] = [0] [a__b] = [4] [b] = [1] [mark](x1) = [4] [f](x1, x2, x3) = [1] x2 + [3] The following symbols are considered usable {a__f, a__b, mark} The order satisfies the following ordering constraints: [a__f(X1, X2, X3)] = [1] X2 + [4] > [1] X2 + [3] = [f(X1, X2, X3)] [a__f(a(), X, X)] = [1] X + [4] ? [8] = [a__f(X, a__b(), b())] [a__b()] = [4] > [0] = [a()] [a__b()] = [4] > [1] = [b()] [mark(a())] = [4] > [0] = [a()] [mark(b())] = [4] >= [4] = [a__b()] [mark(f(X1, X2, X3))] = [4] ? [8] = [a__f(X1, mark(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 YES(O(1),O(n^1)). Strict Trs: { a__f(a(), X, X) -> a__f(X, a__b(), b()) , mark(b()) -> a__b() , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) } Weak Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__b() -> a() , a__b() -> b() , mark(a()) -> a() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(a__f) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [a__f](x1, x2, x3) = [1] x2 + [4] [a] = [0] [a__b] = [0] [b] = [0] [mark](x1) = [4] [f](x1, x2, x3) = [1] x2 + [3] The following symbols are considered usable {a__f, a__b, mark} The order satisfies the following ordering constraints: [a__f(X1, X2, X3)] = [1] X2 + [4] > [1] X2 + [3] = [f(X1, X2, X3)] [a__f(a(), X, X)] = [1] X + [4] >= [4] = [a__f(X, a__b(), b())] [a__b()] = [0] >= [0] = [a()] [a__b()] = [0] >= [0] = [b()] [mark(a())] = [4] > [0] = [a()] [mark(b())] = [4] > [0] = [a__b()] [mark(f(X1, X2, X3))] = [4] ? [8] = [a__f(X1, mark(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 YES(O(1),O(n^1)). Strict Trs: { a__f(a(), X, X) -> a__f(X, a__b(), b()) , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) } Weak Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__b() -> a() , a__b() -> b() , mark(a()) -> a() , mark(b()) -> a__b() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) } 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(a__f) = {2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [a__f](x1, x2, x3) = [1] x2 + [4] [a] = [0] [a__b] = [0] [b] = [0] [mark](x1) = [2] x1 + [0] [f](x1, x2, x3) = [1] x2 + [4] The following symbols are considered usable {a__f, a__b, mark} The order satisfies the following ordering constraints: [a__f(X1, X2, X3)] = [1] X2 + [4] >= [1] X2 + [4] = [f(X1, X2, X3)] [a__f(a(), X, X)] = [1] X + [4] >= [4] = [a__f(X, a__b(), b())] [a__b()] = [0] >= [0] = [a()] [a__b()] = [0] >= [0] = [b()] [mark(a())] = [0] >= [0] = [a()] [mark(b())] = [0] >= [0] = [a__b()] [mark(f(X1, X2, X3))] = [2] X2 + [8] > [2] X2 + [4] = [a__f(X1, mark(X2), X3)] We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict Trs: { a__f(a(), X, X) -> a__f(X, a__b(), b()) } Weak Trs: { a__f(X1, X2, X3) -> f(X1, X2, X3) , a__b() -> a() , a__b() -> b() , mark(a()) -> a() , mark(b()) -> a__b() , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { a__f(a(), X, X) -> a__f(X, a__b(), b()) } 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(a__f) = {2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [a__f](x1, x2, x3) = [4] x1 + [1] x2 + [4] x3 + [0] [a] = [2] [a__b] = [2] [b] = [0] [mark](x1) = [4] x1 + [2] [f](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The following symbols are considered usable {a__f, a__b, mark} The order satisfies the following ordering constraints: [a__f(X1, X2, X3)] = [4] X1 + [1] X2 + [4] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = [f(X1, X2, X3)] [a__f(a(), X, X)] = [5] X + [8] > [4] X + [2] = [a__f(X, a__b(), b())] [a__b()] = [2] >= [2] = [a()] [a__b()] = [2] > [0] = [b()] [mark(a())] = [10] > [2] = [a()] [mark(b())] = [2] >= [2] = [a__b()] [mark(f(X1, X2, X3))] = [4] X1 + [4] X2 + [4] X3 + [2] >= [4] X1 + [4] X2 + [4] X3 + [2] = [a__f(X1, mark(X2), X3)] 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, X3) -> f(X1, X2, X3) , a__f(a(), X, X) -> a__f(X, a__b(), b()) , a__b() -> a() , a__b() -> b() , mark(a()) -> a() , mark(b()) -> a__b() , mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^1))