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: { f(empty(), l) -> l , f(cons(x, k), l) -> g(k, l, cons(x, k)) , g(a, b, c) -> f(a, cons(b, c)) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We add the following weak dependency pairs: Strict DPs: { f^#(empty(), l) -> c_1(l) , f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) , g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { f^#(empty(), l) -> c_1(l) , f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) , g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) } Strict Trs: { f(empty(), l) -> l , f(cons(x, k), l) -> g(k, l, cons(x, k)) , g(a, b, c) -> f(a, cons(b, c)) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { f^#(empty(), l) -> c_1(l) , f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) , g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(c_2) = {1}, Uargs(c_3) = {1} TcT has computed the following constructor-restricted matrix interpretation. [empty] = [0] [0] [cons](x1, x2) = [1 0] x1 + [0] [0 0] [0] [f^#](x1, x2) = [0] [0] [c_1](x1) = [1] [0] [c_2](x1) = [1 0] x1 + [0] [0 1] [0] [g^#](x1, x2, x3) = [0 0] x2 + [1] [1 1] [0] [c_3](x1) = [1 0] x1 + [0] [0 1] [0] The following symbols are considered usable {f^#, g^#} The order satisfies the following ordering constraints: [f^#(empty(), l)] = [0] [0] ? [1] [0] = [c_1(l)] [f^#(cons(x, k), l)] = [0] [0] ? [0 0] l + [1] [1 1] [0] = [c_2(g^#(k, l, cons(x, k)))] [g^#(a, b, c)] = [0 0] b + [1] [1 1] [0] > [0] [0] = [c_3(f^#(a, cons(b, c)))] 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 DPs: { f^#(empty(), l) -> c_1(l) , f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) } Weak DPs: { g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: f^#(empty(), l) -> c_1(l) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_2) = {1}, Uargs(c_3) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [f](x1, x2) = [7] x1 + [7] x2 + [0] [empty] = [4] [cons](x1, x2) = [1] x1 + [1] x2 + [0] [g](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [f^#](x1, x2) = [1] x1 + [0] [c_1](x1) = [3] [c_2](x1) = [1] x1 + [0] [g^#](x1, x2, x3) = [1] x1 + [0] [c_3](x1) = [1] x1 + [0] The following symbols are considered usable {f^#, g^#} The order satisfies the following ordering constraints: [f^#(empty(), l)] = [4] > [3] = [c_1(l)] [f^#(cons(x, k), l)] = [1] x + [1] k + [0] >= [1] k + [0] = [c_2(g^#(k, l, cons(x, k)))] [g^#(a, b, c)] = [1] a + [0] >= [1] a + [0] = [c_3(f^#(a, cons(b, c)))] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) } Weak DPs: { f^#(empty(), l) -> c_1(l) , g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_2) = {1}, Uargs(c_3) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [f](x1, x2) = [7] x1 + [7] x2 + [0] [empty] = [0] [cons](x1, x2) = [1] x1 + [1] x2 + [1] [g](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [f^#](x1, x2) = [1] x1 + [0] [c_1](x1) = [0] [c_2](x1) = [1] x1 + [0] [g^#](x1, x2, x3) = [1] x1 + [0] [c_3](x1) = [1] x1 + [0] The following symbols are considered usable {f^#, g^#} The order satisfies the following ordering constraints: [f^#(empty(), l)] = [0] >= [0] = [c_1(l)] [f^#(cons(x, k), l)] = [1] x + [1] k + [1] > [1] k + [0] = [c_2(g^#(k, l, cons(x, k)))] [g^#(a, b, c)] = [1] a + [0] >= [1] a + [0] = [c_3(f^#(a, cons(b, c)))] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { f^#(empty(), l) -> c_1(l) , f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) , g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) } Obligation: runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { f^#(empty(), l) -> c_1(l) , f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) , g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^1))