YES(O(1),O(n^1)) 0.00/0.59 YES(O(1),O(n^1)) 0.00/0.59 0.00/0.59 We are left with following problem, upon which TcT provides the 0.00/0.59 certificate YES(O(1),O(n^1)). 0.00/0.59 0.00/0.59 Strict Trs: 0.00/0.59 { from(X) -> cons(X, n__from(s(X))) 0.00/0.59 , from(X) -> n__from(X) 0.00/0.59 , after(s(N), cons(X, XS)) -> after(N, activate(XS)) 0.00/0.59 , after(0(), XS) -> XS 0.00/0.59 , activate(X) -> X 0.00/0.59 , activate(n__from(X)) -> from(X) } 0.00/0.59 Obligation: 0.00/0.59 runtime complexity 0.00/0.59 Answer: 0.00/0.59 YES(O(1),O(n^1)) 0.00/0.59 0.00/0.59 We add the following weak dependency pairs: 0.00/0.59 0.00/0.59 Strict DPs: 0.00/0.59 { from^#(X) -> c_1(X, X) 0.00/0.59 , from^#(X) -> c_2(X) 0.00/0.59 , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) 0.00/0.59 , after^#(0(), XS) -> c_4(XS) 0.00/0.59 , activate^#(X) -> c_5(X) 0.00/0.59 , activate^#(n__from(X)) -> c_6(from^#(X)) } 0.00/0.59 0.00/0.59 and mark the set of starting terms. 0.00/0.59 0.00/0.59 We are left with following problem, upon which TcT provides the 0.00/0.59 certificate YES(O(1),O(n^1)). 0.00/0.59 0.00/0.59 Strict DPs: 0.00/0.59 { from^#(X) -> c_1(X, X) 0.00/0.59 , from^#(X) -> c_2(X) 0.00/0.59 , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) 0.00/0.59 , after^#(0(), XS) -> c_4(XS) 0.00/0.59 , activate^#(X) -> c_5(X) 0.00/0.59 , activate^#(n__from(X)) -> c_6(from^#(X)) } 0.00/0.59 Strict Trs: 0.00/0.59 { from(X) -> cons(X, n__from(s(X))) 0.00/0.59 , from(X) -> n__from(X) 0.00/0.59 , after(s(N), cons(X, XS)) -> after(N, activate(XS)) 0.00/0.59 , after(0(), XS) -> XS 0.00/0.59 , activate(X) -> X 0.00/0.59 , activate(n__from(X)) -> from(X) } 0.00/0.59 Obligation: 0.00/0.59 runtime complexity 0.00/0.59 Answer: 0.00/0.59 YES(O(1),O(n^1)) 0.00/0.59 0.00/0.59 We replace rewrite rules by usable rules: 0.00/0.59 0.00/0.59 Strict Usable Rules: 0.00/0.59 { from(X) -> cons(X, n__from(s(X))) 0.00/0.59 , from(X) -> n__from(X) 0.00/0.59 , activate(X) -> X 0.00/0.59 , activate(n__from(X)) -> from(X) } 0.00/0.59 0.00/0.59 We are left with following problem, upon which TcT provides the 0.00/0.59 certificate YES(O(1),O(n^1)). 0.00/0.59 0.00/0.59 Strict DPs: 0.00/0.59 { from^#(X) -> c_1(X, X) 0.00/0.59 , from^#(X) -> c_2(X) 0.00/0.59 , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) 0.00/0.59 , after^#(0(), XS) -> c_4(XS) 0.00/0.59 , activate^#(X) -> c_5(X) 0.00/0.59 , activate^#(n__from(X)) -> c_6(from^#(X)) } 0.00/0.59 Strict Trs: 0.00/0.59 { from(X) -> cons(X, n__from(s(X))) 0.00/0.59 , from(X) -> n__from(X) 0.00/0.59 , activate(X) -> X 0.00/0.59 , activate(n__from(X)) -> from(X) } 0.00/0.59 Obligation: 0.00/0.59 runtime complexity 0.00/0.59 Answer: 0.00/0.59 YES(O(1),O(n^1)) 0.00/0.59 0.00/0.59 The weightgap principle applies (using the following constant 0.00/0.59 growth matrix-interpretation) 0.00/0.59 0.00/0.59 The following argument positions are usable: 0.00/0.59 Uargs(after^#) = {2}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, 0.00/0.59 Uargs(c_6) = {1} 0.00/0.59 0.00/0.59 TcT has computed the following constructor-restricted matrix 0.00/0.59 interpretation. 0.00/0.59 0.00/0.59 [from](x1) = [1] 0.00/0.59 [0] 0.00/0.59 0.00/0.59 [cons](x1, x2) = [1 0] x2 + [0] 0.00/0.59 [0 0] [0] 0.00/0.59 0.00/0.59 [n__from](x1) = [0] 0.00/0.59 [0] 0.00/0.59 0.00/0.59 [s](x1) = [1 2] x1 + [0] 0.00/0.59 [0 0] [0] 0.00/0.59 0.00/0.59 [0] = [0] 0.00/0.59 [0] 0.00/0.59 0.00/0.59 [activate](x1) = [1 0] x1 + [2] 0.00/0.59 [0 2] [0] 0.00/0.59 0.00/0.59 [from^#](x1) = [0 0] x1 + [1] 0.00/0.59 [2 1] [1] 0.00/0.59 0.00/0.59 [c_1](x1, x2) = [0 0] x1 + [0 0] x2 + [1] 0.00/0.59 [1 2] [2 1] [1] 0.00/0.59 0.00/0.59 [c_2](x1) = [0 0] x1 + [0] 0.00/0.59 [1 1] [1] 0.00/0.59 0.00/0.59 [after^#](x1, x2) = [1 0] x2 + [0] 0.00/0.59 [0 0] [0] 0.00/0.59 0.00/0.59 [c_3](x1) = [1 0] x1 + [2] 0.00/0.59 [0 1] [2] 0.00/0.59 0.00/0.59 [c_4](x1) = [1 0] x1 + [1] 0.00/0.59 [0 1] [0] 0.00/0.59 0.00/0.59 [activate^#](x1) = [1 1] x1 + [2] 0.00/0.59 [1 1] [2] 0.00/0.59 0.00/0.59 [c_5](x1) = [1 1] x1 + [1] 0.00/0.59 [1 1] [1] 0.00/0.59 0.00/0.59 [c_6](x1) = [1 0] x1 + [2] 0.00/0.59 [0 1] [2] 0.00/0.59 0.00/0.59 The order satisfies the following ordering constraints: 0.00/0.59 0.00/0.59 [from(X)] = [1] 0.00/0.59 [0] 0.00/0.59 > [0] 0.00/0.59 [0] 0.00/0.59 = [cons(X, n__from(s(X)))] 0.00/0.59 0.00/0.59 [from(X)] = [1] 0.00/0.59 [0] 0.00/0.59 > [0] 0.00/0.59 [0] 0.00/0.59 = [n__from(X)] 0.00/0.59 0.00/0.59 [activate(X)] = [1 0] X + [2] 0.00/0.59 [0 2] [0] 0.00/0.59 > [1 0] X + [0] 0.00/0.59 [0 1] [0] 0.00/0.59 = [X] 0.00/0.59 0.00/0.59 [activate(n__from(X))] = [2] 0.00/0.59 [0] 0.00/0.59 > [1] 0.00/0.59 [0] 0.00/0.59 = [from(X)] 0.00/0.59 0.00/0.59 [from^#(X)] = [0 0] X + [1] 0.00/0.59 [2 1] [1] 0.00/0.59 ? [0 0] X + [1] 0.00/0.59 [3 3] [1] 0.00/0.59 = [c_1(X, X)] 0.00/0.59 0.00/0.59 [from^#(X)] = [0 0] X + [1] 0.00/0.59 [2 1] [1] 0.00/0.59 > [0 0] X + [0] 0.00/0.59 [1 1] [1] 0.00/0.59 = [c_2(X)] 0.00/0.59 0.00/0.59 [after^#(s(N), cons(X, XS))] = [1 0] XS + [0] 0.00/0.59 [0 0] [0] 0.00/0.59 ? [1 0] XS + [4] 0.00/0.59 [0 0] [2] 0.00/0.59 = [c_3(after^#(N, activate(XS)))] 0.00/0.59 0.00/0.59 [after^#(0(), XS)] = [1 0] XS + [0] 0.00/0.59 [0 0] [0] 0.00/0.59 ? [1 0] XS + [1] 0.00/0.59 [0 1] [0] 0.00/0.59 = [c_4(XS)] 0.00/0.59 0.00/0.59 [activate^#(X)] = [1 1] X + [2] 0.00/0.59 [1 1] [2] 0.00/0.59 > [1 1] X + [1] 0.00/0.59 [1 1] [1] 0.00/0.59 = [c_5(X)] 0.00/0.59 0.00/0.59 [activate^#(n__from(X))] = [2] 0.00/0.59 [2] 0.00/0.59 ? [0 0] X + [3] 0.00/0.59 [2 1] [3] 0.00/0.59 = [c_6(from^#(X))] 0.00/0.59 0.00/0.59 0.00/0.59 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.59 0.00/0.59 We are left with following problem, upon which TcT provides the 0.00/0.59 certificate YES(O(1),O(n^1)). 0.00/0.59 0.00/0.59 Strict DPs: 0.00/0.59 { from^#(X) -> c_1(X, X) 0.00/0.59 , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) 0.00/0.59 , after^#(0(), XS) -> c_4(XS) 0.00/0.59 , activate^#(n__from(X)) -> c_6(from^#(X)) } 0.00/0.59 Weak DPs: 0.00/0.59 { from^#(X) -> c_2(X) 0.00/0.59 , activate^#(X) -> c_5(X) } 0.00/0.59 Weak Trs: 0.00/0.59 { from(X) -> cons(X, n__from(s(X))) 0.00/0.59 , from(X) -> n__from(X) 0.00/0.59 , activate(X) -> X 0.00/0.59 , activate(n__from(X)) -> from(X) } 0.00/0.59 Obligation: 0.00/0.59 runtime complexity 0.00/0.59 Answer: 0.00/0.59 YES(O(1),O(n^1)) 0.00/0.59 0.00/0.59 We use the processor 'matrix interpretation of dimension 1' to 0.00/0.59 orient following rules strictly. 0.00/0.59 0.00/0.59 DPs: 0.00/0.59 { 2: after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) 0.00/0.59 , 3: after^#(0(), XS) -> c_4(XS) 0.00/0.59 , 4: activate^#(n__from(X)) -> c_6(from^#(X)) 0.00/0.59 , 6: activate^#(X) -> c_5(X) } 0.00/0.59 0.00/0.59 Sub-proof: 0.00/0.59 ---------- 0.00/0.59 The following argument positions are usable: 0.00/0.59 Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1} 0.00/0.59 0.00/0.59 TcT has computed the following constructor-based matrix 0.00/0.59 interpretation satisfying not(EDA). 0.00/0.59 0.00/0.59 [from](x1) = [1] 0.00/0.59 0.00/0.59 [cons](x1, x2) = [1] x2 + [0] 0.00/0.59 0.00/0.59 [n__from](x1) = [1] 0.00/0.59 0.00/0.59 [s](x1) = [1] x1 + [1] 0.00/0.59 0.00/0.59 [0] = [0] 0.00/0.59 0.00/0.59 [activate](x1) = [1] x1 + [0] 0.00/0.59 0.00/0.59 [from^#](x1) = [0] 0.00/0.59 0.00/0.59 [c_1](x1, x2) = [0] 0.00/0.59 0.00/0.59 [c_2](x1) = [0] 0.00/0.59 0.00/0.59 [after^#](x1, x2) = [7] x1 + [4] x2 + [1] 0.00/0.59 0.00/0.59 [c_3](x1) = [1] x1 + [1] 0.00/0.59 0.00/0.59 [c_4](x1) = [4] x1 + [0] 0.00/0.59 0.00/0.59 [activate^#](x1) = [1] x1 + [7] 0.00/0.59 0.00/0.59 [c_5](x1) = [1] x1 + [6] 0.00/0.59 0.00/0.59 [c_6](x1) = [4] x1 + [7] 0.00/0.59 0.00/0.59 The order satisfies the following ordering constraints: 0.00/0.59 0.00/0.59 [from(X)] = [1] 0.00/0.59 >= [1] 0.00/0.59 = [cons(X, n__from(s(X)))] 0.00/0.59 0.00/0.59 [from(X)] = [1] 0.00/0.59 >= [1] 0.00/0.59 = [n__from(X)] 0.00/0.59 0.00/0.59 [activate(X)] = [1] X + [0] 0.00/0.59 >= [1] X + [0] 0.00/0.59 = [X] 0.00/0.59 0.00/0.59 [activate(n__from(X))] = [1] 0.00/0.59 >= [1] 0.00/0.59 = [from(X)] 0.00/0.59 0.00/0.59 [from^#(X)] = [0] 0.00/0.59 >= [0] 0.00/0.59 = [c_1(X, X)] 0.00/0.59 0.00/0.59 [from^#(X)] = [0] 0.00/0.59 >= [0] 0.00/0.59 = [c_2(X)] 0.00/0.59 0.00/0.59 [after^#(s(N), cons(X, XS))] = [4] XS + [7] N + [8] 0.00/0.59 > [4] XS + [7] N + [2] 0.00/0.59 = [c_3(after^#(N, activate(XS)))] 0.00/0.59 0.00/0.59 [after^#(0(), XS)] = [4] XS + [1] 0.00/0.59 > [4] XS + [0] 0.00/0.59 = [c_4(XS)] 0.00/0.59 0.00/0.59 [activate^#(X)] = [1] X + [7] 0.00/0.59 > [1] X + [6] 0.00/0.59 = [c_5(X)] 0.00/0.59 0.00/0.59 [activate^#(n__from(X))] = [8] 0.00/0.59 > [7] 0.00/0.59 = [c_6(from^#(X))] 0.00/0.59 0.00/0.59 0.00/0.59 The strictly oriented rules are moved into the weak component. 0.00/0.59 0.00/0.59 We are left with following problem, upon which TcT provides the 0.00/0.59 certificate YES(O(1),O(n^1)). 0.00/0.59 0.00/0.59 Strict DPs: { from^#(X) -> c_1(X, X) } 0.00/0.59 Weak DPs: 0.00/0.59 { from^#(X) -> c_2(X) 0.00/0.59 , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) 0.00/0.59 , after^#(0(), XS) -> c_4(XS) 0.00/0.59 , activate^#(X) -> c_5(X) 0.00/0.59 , activate^#(n__from(X)) -> c_6(from^#(X)) } 0.00/0.59 Weak Trs: 0.00/0.59 { from(X) -> cons(X, n__from(s(X))) 0.00/0.59 , from(X) -> n__from(X) 0.00/0.59 , activate(X) -> X 0.00/0.59 , activate(n__from(X)) -> from(X) } 0.00/0.59 Obligation: 0.00/0.59 runtime complexity 0.00/0.59 Answer: 0.00/0.59 YES(O(1),O(n^1)) 0.00/0.59 0.00/0.59 We use the processor 'matrix interpretation of dimension 1' to 0.00/0.59 orient following rules strictly. 0.00/0.59 0.00/0.59 DPs: 0.00/0.59 { 1: from^#(X) -> c_1(X, X) 0.00/0.59 , 2: from^#(X) -> c_2(X) 0.00/0.59 , 4: after^#(0(), XS) -> c_4(XS) 0.00/0.59 , 5: activate^#(X) -> c_5(X) 0.00/0.59 , 6: activate^#(n__from(X)) -> c_6(from^#(X)) } 0.00/0.59 Trs: 0.00/0.59 { activate(X) -> X 0.00/0.59 , activate(n__from(X)) -> from(X) } 0.00/0.59 0.00/0.59 Sub-proof: 0.00/0.59 ---------- 0.00/0.59 The following argument positions are usable: 0.00/0.59 Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1} 0.00/0.59 0.00/0.59 TcT has computed the following constructor-based matrix 0.00/0.59 interpretation satisfying not(EDA). 0.00/0.59 0.00/0.59 [from](x1) = [0] 0.00/0.59 0.00/0.59 [cons](x1, x2) = [1] x2 + [0] 0.00/0.59 0.00/0.59 [n__from](x1) = [0] 0.00/0.59 0.00/0.59 [s](x1) = [1] x1 + [1] 0.00/0.59 0.00/0.59 [0] = [2] 0.00/0.59 0.00/0.59 [activate](x1) = [1] x1 + [4] 0.00/0.59 0.00/0.59 [from^#](x1) = [1] 0.00/0.59 0.00/0.59 [c_1](x1, x2) = [0] 0.00/0.59 0.00/0.59 [c_2](x1) = [0] 0.00/0.59 0.00/0.59 [after^#](x1, x2) = [4] x1 + [1] x2 + [0] 0.00/0.59 0.00/0.59 [c_3](x1) = [1] x1 + [0] 0.00/0.59 0.00/0.59 [c_4](x1) = [1] x1 + [6] 0.00/0.59 0.00/0.59 [activate^#](x1) = [7] x1 + [7] 0.00/0.59 0.00/0.59 [c_5](x1) = [7] x1 + [6] 0.00/0.59 0.00/0.59 [c_6](x1) = [4] x1 + [0] 0.00/0.59 0.00/0.59 The order satisfies the following ordering constraints: 0.00/0.59 0.00/0.59 [from(X)] = [0] 0.00/0.59 >= [0] 0.00/0.59 = [cons(X, n__from(s(X)))] 0.00/0.59 0.00/0.59 [from(X)] = [0] 0.00/0.59 >= [0] 0.00/0.59 = [n__from(X)] 0.00/0.59 0.00/0.59 [activate(X)] = [1] X + [4] 0.00/0.59 > [1] X + [0] 0.00/0.59 = [X] 0.00/0.59 0.00/0.59 [activate(n__from(X))] = [4] 0.00/0.59 > [0] 0.00/0.59 = [from(X)] 0.00/0.59 0.00/0.59 [from^#(X)] = [1] 0.00/0.59 > [0] 0.00/0.59 = [c_1(X, X)] 0.00/0.59 0.00/0.59 [from^#(X)] = [1] 0.00/0.59 > [0] 0.00/0.59 = [c_2(X)] 0.00/0.59 0.00/0.59 [after^#(s(N), cons(X, XS))] = [1] XS + [4] N + [4] 0.00/0.59 >= [1] XS + [4] N + [4] 0.00/0.59 = [c_3(after^#(N, activate(XS)))] 0.00/0.59 0.00/0.59 [after^#(0(), XS)] = [1] XS + [8] 0.00/0.59 > [1] XS + [6] 0.00/0.59 = [c_4(XS)] 0.00/0.59 0.00/0.59 [activate^#(X)] = [7] X + [7] 0.00/0.59 > [7] X + [6] 0.00/0.59 = [c_5(X)] 0.00/0.59 0.00/0.59 [activate^#(n__from(X))] = [7] 0.00/0.59 > [4] 0.00/0.59 = [c_6(from^#(X))] 0.00/0.59 0.00/0.59 0.00/0.59 The strictly oriented rules are moved into the weak component. 0.00/0.59 0.00/0.59 We are left with following problem, upon which TcT provides the 0.00/0.59 certificate YES(O(1),O(1)). 0.00/0.59 0.00/0.59 Weak DPs: 0.00/0.59 { from^#(X) -> c_1(X, X) 0.00/0.59 , from^#(X) -> c_2(X) 0.00/0.59 , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) 0.00/0.59 , after^#(0(), XS) -> c_4(XS) 0.00/0.59 , activate^#(X) -> c_5(X) 0.00/0.59 , activate^#(n__from(X)) -> c_6(from^#(X)) } 0.00/0.59 Weak Trs: 0.00/0.59 { from(X) -> cons(X, n__from(s(X))) 0.00/0.59 , from(X) -> n__from(X) 0.00/0.59 , activate(X) -> X 0.00/0.59 , activate(n__from(X)) -> from(X) } 0.00/0.59 Obligation: 0.00/0.59 runtime complexity 0.00/0.59 Answer: 0.00/0.59 YES(O(1),O(1)) 0.00/0.59 0.00/0.59 The following weak DPs constitute a sub-graph of the DG that is 0.00/0.59 closed under successors. The DPs are removed. 0.00/0.59 0.00/0.59 { from^#(X) -> c_1(X, X) 0.00/0.59 , from^#(X) -> c_2(X) 0.00/0.59 , after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS))) 0.00/0.59 , after^#(0(), XS) -> c_4(XS) 0.00/0.59 , activate^#(X) -> c_5(X) 0.00/0.59 , activate^#(n__from(X)) -> c_6(from^#(X)) } 0.00/0.59 0.00/0.59 We are left with following problem, upon which TcT provides the 0.00/0.59 certificate YES(O(1),O(1)). 0.00/0.59 0.00/0.59 Weak Trs: 0.00/0.59 { from(X) -> cons(X, n__from(s(X))) 0.00/0.59 , from(X) -> n__from(X) 0.00/0.59 , activate(X) -> X 0.00/0.59 , activate(n__from(X)) -> from(X) } 0.00/0.59 Obligation: 0.00/0.59 runtime complexity 0.00/0.59 Answer: 0.00/0.59 YES(O(1),O(1)) 0.00/0.59 0.00/0.59 No rule is usable, rules are removed from the input problem. 0.00/0.59 0.00/0.59 We are left with following problem, upon which TcT provides the 0.00/0.59 certificate YES(O(1),O(1)). 0.00/0.59 0.00/0.59 Rules: Empty 0.00/0.59 Obligation: 0.00/0.59 runtime complexity 0.00/0.59 Answer: 0.00/0.59 YES(O(1),O(1)) 0.00/0.59 0.00/0.59 Empty rules are trivially bounded 0.00/0.59 0.00/0.59 Hurray, we answered YES(O(1),O(n^1)) 0.00/0.59 EOF