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: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , after(s(N), cons(X, XS)) -> after(N, activate(XS)) , after(0(), XS) -> XS , activate(X) -> X , activate(n__from(X)) -> from(X) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We add the following weak dependency pairs: Strict DPs: { from^#(X) -> c_1(X, X) , from^#(X) -> c_2(X) , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) , after^#(0(), XS) -> c_4(XS) , activate^#(X) -> c_5(X) , activate^#(n__from(X)) -> c_6(from^#(X)) } 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: { from^#(X) -> c_1(X, X) , from^#(X) -> c_2(X) , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) , after^#(0(), XS) -> c_4(XS) , activate^#(X) -> c_5(X) , activate^#(n__from(X)) -> c_6(from^#(X)) } Strict Trs: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , after(s(N), cons(X, XS)) -> after(N, activate(XS)) , after(0(), XS) -> XS , activate(X) -> X , activate(n__from(X)) -> from(X) } Obligation: runtime complexity Answer: YES(O(1),O(n^1)) We replace rewrite rules by usable rules: Strict Usable Rules: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , activate(X) -> X , activate(n__from(X)) -> from(X) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { from^#(X) -> c_1(X, X) , from^#(X) -> c_2(X) , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) , after^#(0(), XS) -> c_4(XS) , activate^#(X) -> c_5(X) , activate^#(n__from(X)) -> c_6(from^#(X)) } Strict Trs: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , activate(X) -> X , activate(n__from(X)) -> from(X) } 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(after^#) = {2}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1} TcT has computed the following constructor-restricted matrix interpretation. [from](x1) = [1] [0] [cons](x1, x2) = [1 0] x2 + [0] [0 0] [0] [n__from](x1) = [0] [0] [s](x1) = [1 2] x1 + [0] [0 0] [0] [0] = [0] [0] [activate](x1) = [1 0] x1 + [2] [0 2] [0] [from^#](x1) = [0 0] x1 + [1] [2 1] [1] [c_1](x1, x2) = [0 0] x1 + [0 0] x2 + [1] [1 2] [2 1] [1] [c_2](x1) = [0 0] x1 + [0] [1 1] [1] [after^#](x1, x2) = [1 0] x2 + [0] [0 0] [0] [c_3](x1) = [1 0] x1 + [2] [0 1] [2] [c_4](x1) = [1 0] x1 + [1] [0 1] [0] [activate^#](x1) = [1 1] x1 + [2] [1 1] [2] [c_5](x1) = [1 1] x1 + [1] [1 1] [1] [c_6](x1) = [1 0] x1 + [2] [0 1] [2] The following symbols are considered usable {from, activate, from^#, after^#, activate^#} The order satisfies the following ordering constraints: [from(X)] = [1] [0] > [0] [0] = [cons(X, n__from(s(X)))] [from(X)] = [1] [0] > [0] [0] = [n__from(X)] [activate(X)] = [1 0] X + [2] [0 2] [0] > [1 0] X + [0] [0 1] [0] = [X] [activate(n__from(X))] = [2] [0] > [1] [0] = [from(X)] [from^#(X)] = [0 0] X + [1] [2 1] [1] ? [0 0] X + [1] [3 3] [1] = [c_1(X, X)] [from^#(X)] = [0 0] X + [1] [2 1] [1] > [0 0] X + [0] [1 1] [1] = [c_2(X)] [after^#(s(N), cons(X, XS))] = [1 0] XS + [0] [0 0] [0] ? [1 0] XS + [4] [0 0] [2] = [c_3(after^#(N, activate(XS)))] [after^#(0(), XS)] = [1 0] XS + [0] [0 0] [0] ? [1 0] XS + [1] [0 1] [0] = [c_4(XS)] [activate^#(X)] = [1 1] X + [2] [1 1] [2] > [1 1] X + [1] [1 1] [1] = [c_5(X)] [activate^#(n__from(X))] = [2] [2] ? [0 0] X + [3] [2 1] [3] = [c_6(from^#(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 YES(O(1),O(n^1)). Strict DPs: { from^#(X) -> c_1(X, X) , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) , after^#(0(), XS) -> c_4(XS) , activate^#(n__from(X)) -> c_6(from^#(X)) } Weak DPs: { from^#(X) -> c_2(X) , activate^#(X) -> c_5(X) } Weak Trs: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , activate(X) -> X , activate(n__from(X)) -> from(X) } 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: from^#(X) -> c_1(X, X) , 4: activate^#(n__from(X)) -> c_6(from^#(X)) , 5: from^#(X) -> c_2(X) , 6: activate^#(X) -> c_5(X) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [from](x1) = [1] x1 + [4] [cons](x1, x2) = [1] x2 + [0] [n__from](x1) = [1] x1 + [4] [s](x1) = [1] x1 + [0] [after](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [7] [activate](x1) = [1] x1 + [0] [from^#](x1) = [1] x1 + [4] [c_1](x1, x2) = [1] x2 + [3] [c_2](x1) = [1] x1 + [3] [after^#](x1, x2) = [1] x2 + [0] [c_3](x1) = [1] x1 + [0] [c_4](x1) = [1] x1 + [0] [activate^#](x1) = [2] x1 + [7] [c_5](x1) = [1] x1 + [5] [c_6](x1) = [2] x1 + [3] The following symbols are considered usable {from, activate, from^#, after^#, activate^#} The order satisfies the following ordering constraints: [from(X)] = [1] X + [4] >= [1] X + [4] = [cons(X, n__from(s(X)))] [from(X)] = [1] X + [4] >= [1] X + [4] = [n__from(X)] [activate(X)] = [1] X + [0] >= [1] X + [0] = [X] [activate(n__from(X))] = [1] X + [4] >= [1] X + [4] = [from(X)] [from^#(X)] = [1] X + [4] > [1] X + [3] = [c_1(X, X)] [from^#(X)] = [1] X + [4] > [1] X + [3] = [c_2(X)] [after^#(s(N), cons(X, XS))] = [1] XS + [0] >= [1] XS + [0] = [c_3(after^#(N, activate(XS)))] [after^#(0(), XS)] = [1] XS + [0] >= [1] XS + [0] = [c_4(XS)] [activate^#(X)] = [2] X + [7] > [1] X + [5] = [c_5(X)] [activate^#(n__from(X))] = [2] X + [15] > [2] X + [11] = [c_6(from^#(X))] 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: { after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) , after^#(0(), XS) -> c_4(XS) } Weak DPs: { from^#(X) -> c_1(X, X) , from^#(X) -> c_2(X) , activate^#(X) -> c_5(X) , activate^#(n__from(X)) -> c_6(from^#(X)) } Weak Trs: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , activate(X) -> X , activate(n__from(X)) -> from(X) } 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: { 2: after^#(0(), XS) -> c_4(XS) , 5: activate^#(X) -> c_5(X) , 6: activate^#(n__from(X)) -> c_6(from^#(X)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [from](x1) = [1] x1 + [0] [cons](x1, x2) = [1] x2 + [0] [n__from](x1) = [1] x1 + [0] [s](x1) = [1] x1 + [0] [after](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [1] [activate](x1) = [1] x1 + [0] [from^#](x1) = [1] x1 + [0] [c_1](x1, x2) = [1] x1 + [0] [c_2](x1) = [1] x1 + [0] [after^#](x1, x2) = [1] x1 + [2] x2 + [0] [c_3](x1) = [1] x1 + [0] [c_4](x1) = [2] x1 + [0] [activate^#](x1) = [7] x1 + [7] [c_5](x1) = [7] x1 + [6] [c_6](x1) = [4] x1 + [1] The following symbols are considered usable {from, activate, from^#, after^#, activate^#} The order satisfies the following ordering constraints: [from(X)] = [1] X + [0] >= [1] X + [0] = [cons(X, n__from(s(X)))] [from(X)] = [1] X + [0] >= [1] X + [0] = [n__from(X)] [activate(X)] = [1] X + [0] >= [1] X + [0] = [X] [activate(n__from(X))] = [1] X + [0] >= [1] X + [0] = [from(X)] [from^#(X)] = [1] X + [0] >= [1] X + [0] = [c_1(X, X)] [from^#(X)] = [1] X + [0] >= [1] X + [0] = [c_2(X)] [after^#(s(N), cons(X, XS))] = [2] XS + [1] N + [0] >= [2] XS + [1] N + [0] = [c_3(after^#(N, activate(XS)))] [after^#(0(), XS)] = [2] XS + [1] > [2] XS + [0] = [c_4(XS)] [activate^#(X)] = [7] X + [7] > [7] X + [6] = [c_5(X)] [activate^#(n__from(X))] = [7] X + [7] > [4] X + [1] = [c_6(from^#(X))] 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: { after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) } Weak DPs: { from^#(X) -> c_1(X, X) , from^#(X) -> c_2(X) , after^#(0(), XS) -> c_4(XS) , activate^#(X) -> c_5(X) , activate^#(n__from(X)) -> c_6(from^#(X)) } Weak Trs: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , activate(X) -> X , activate(n__from(X)) -> from(X) } 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: after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) , 4: after^#(0(), XS) -> c_4(XS) , 5: activate^#(X) -> c_5(X) , 6: activate^#(n__from(X)) -> c_6(from^#(X)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [from](x1) = [0] [cons](x1, x2) = [1] x2 + [0] [n__from](x1) = [0] [s](x1) = [1] x1 + [2] [after](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [2] [activate](x1) = [1] x1 + [0] [from^#](x1) = [0] [c_1](x1, x2) = [0] [c_2](x1) = [0] [after^#](x1, x2) = [4] x1 + [1] x2 + [0] [c_3](x1) = [1] x1 + [1] [c_4](x1) = [1] x1 + [7] [activate^#](x1) = [5] x1 + [7] [c_5](x1) = [5] x1 + [6] [c_6](x1) = [2] x1 + [1] The following symbols are considered usable {from, activate, from^#, after^#, activate^#} The order satisfies the following ordering constraints: [from(X)] = [0] >= [0] = [cons(X, n__from(s(X)))] [from(X)] = [0] >= [0] = [n__from(X)] [activate(X)] = [1] X + [0] >= [1] X + [0] = [X] [activate(n__from(X))] = [0] >= [0] = [from(X)] [from^#(X)] = [0] >= [0] = [c_1(X, X)] [from^#(X)] = [0] >= [0] = [c_2(X)] [after^#(s(N), cons(X, XS))] = [1] XS + [4] N + [8] > [1] XS + [4] N + [1] = [c_3(after^#(N, activate(XS)))] [after^#(0(), XS)] = [1] XS + [8] > [1] XS + [7] = [c_4(XS)] [activate^#(X)] = [5] X + [7] > [5] X + [6] = [c_5(X)] [activate^#(n__from(X))] = [7] > [1] = [c_6(from^#(X))] 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: { from^#(X) -> c_1(X, X) , from^#(X) -> c_2(X) , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) , after^#(0(), XS) -> c_4(XS) , activate^#(X) -> c_5(X) , activate^#(n__from(X)) -> c_6(from^#(X)) } Weak Trs: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , activate(X) -> X , activate(n__from(X)) -> from(X) } 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. { from^#(X) -> c_1(X, X) , from^#(X) -> c_2(X) , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) , after^#(0(), XS) -> c_4(XS) , activate^#(X) -> c_5(X) , activate^#(n__from(X)) -> c_6(from^#(X)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { from(X) -> cons(X, n__from(s(X))) , from(X) -> n__from(X) , activate(X) -> X , activate(n__from(X)) -> from(X) } Obligation: runtime complexity Answer: YES(O(1),O(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(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))