YES(O(1),O(n^1)) 0.00/0.74 YES(O(1),O(n^1)) 0.00/0.74 0.00/0.74 We are left with following problem, upon which TcT provides the 0.00/0.74 certificate YES(O(1),O(n^1)). 0.00/0.74 0.00/0.74 Strict Trs: 0.00/0.74 { from(X) -> cons(X, n__from(n__s(X))) 0.00/0.74 , from(X) -> n__from(X) 0.00/0.74 , first(X1, X2) -> n__first(X1, X2) 0.00/0.74 , first(0(), Z) -> nil() 0.00/0.74 , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) 0.00/0.74 , s(X) -> n__s(X) 0.00/0.74 , activate(X) -> X 0.00/0.74 , activate(n__from(X)) -> from(activate(X)) 0.00/0.74 , activate(n__s(X)) -> s(activate(X)) 0.00/0.74 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 0.00/0.74 , sel(0(), cons(X, Z)) -> X 0.00/0.74 , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) } 0.00/0.74 Obligation: 0.00/0.74 innermost runtime complexity 0.00/0.74 Answer: 0.00/0.74 YES(O(1),O(n^1)) 0.00/0.74 0.00/0.74 Arguments of following rules are not normal-forms: 0.00/0.74 0.00/0.74 { first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) 0.00/0.74 , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) } 0.00/0.74 0.00/0.74 All above mentioned rules can be savely removed. 0.00/0.74 0.00/0.74 We are left with following problem, upon which TcT provides the 0.00/0.74 certificate YES(O(1),O(n^1)). 0.00/0.74 0.00/0.74 Strict Trs: 0.00/0.74 { from(X) -> cons(X, n__from(n__s(X))) 0.00/0.74 , from(X) -> n__from(X) 0.00/0.74 , first(X1, X2) -> n__first(X1, X2) 0.00/0.74 , first(0(), Z) -> nil() 0.00/0.74 , s(X) -> n__s(X) 0.00/0.74 , activate(X) -> X 0.00/0.74 , activate(n__from(X)) -> from(activate(X)) 0.00/0.74 , activate(n__s(X)) -> s(activate(X)) 0.00/0.74 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 0.00/0.74 , sel(0(), cons(X, Z)) -> X } 0.00/0.74 Obligation: 0.00/0.74 innermost runtime complexity 0.00/0.74 Answer: 0.00/0.74 YES(O(1),O(n^1)) 0.00/0.74 0.00/0.74 We add the following weak dependency pairs: 0.00/0.74 0.00/0.74 Strict DPs: 0.00/0.74 { from^#(X) -> c_1() 0.00/0.74 , from^#(X) -> c_2() 0.00/0.74 , first^#(X1, X2) -> c_3() 0.00/0.74 , first^#(0(), Z) -> c_4() 0.00/0.74 , s^#(X) -> c_5() 0.00/0.74 , activate^#(X) -> c_6() 0.00/0.74 , activate^#(n__from(X)) -> c_7(from^#(activate(X))) 0.00/0.74 , activate^#(n__s(X)) -> c_8(s^#(activate(X))) 0.00/0.74 , activate^#(n__first(X1, X2)) -> 0.00/0.74 c_9(first^#(activate(X1), activate(X2))) 0.00/0.74 , sel^#(0(), cons(X, Z)) -> c_10() } 0.00/0.74 0.00/0.74 and mark the set of starting terms. 0.00/0.74 0.00/0.74 We are left with following problem, upon which TcT provides the 0.00/0.74 certificate YES(O(1),O(n^1)). 0.00/0.74 0.00/0.74 Strict DPs: 0.00/0.74 { from^#(X) -> c_1() 0.00/0.74 , from^#(X) -> c_2() 0.00/0.74 , first^#(X1, X2) -> c_3() 0.00/0.74 , first^#(0(), Z) -> c_4() 0.00/0.74 , s^#(X) -> c_5() 0.00/0.74 , activate^#(X) -> c_6() 0.00/0.74 , activate^#(n__from(X)) -> c_7(from^#(activate(X))) 0.00/0.74 , activate^#(n__s(X)) -> c_8(s^#(activate(X))) 0.00/0.74 , activate^#(n__first(X1, X2)) -> 0.00/0.74 c_9(first^#(activate(X1), activate(X2))) 0.00/0.74 , sel^#(0(), cons(X, Z)) -> c_10() } 0.00/0.74 Strict Trs: 0.00/0.74 { from(X) -> cons(X, n__from(n__s(X))) 0.00/0.74 , from(X) -> n__from(X) 0.00/0.74 , first(X1, X2) -> n__first(X1, X2) 0.00/0.74 , first(0(), Z) -> nil() 0.00/0.74 , s(X) -> n__s(X) 0.00/0.74 , activate(X) -> X 0.00/0.74 , activate(n__from(X)) -> from(activate(X)) 0.00/0.74 , activate(n__s(X)) -> s(activate(X)) 0.00/0.74 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 0.00/0.74 , sel(0(), cons(X, Z)) -> X } 0.00/0.74 Obligation: 0.00/0.74 innermost runtime complexity 0.00/0.74 Answer: 0.00/0.74 YES(O(1),O(n^1)) 0.00/0.74 0.00/0.74 We replace rewrite rules by usable rules: 0.00/0.74 0.00/0.74 Strict Usable Rules: 0.00/0.74 { from(X) -> cons(X, n__from(n__s(X))) 0.00/0.74 , from(X) -> n__from(X) 0.00/0.74 , first(X1, X2) -> n__first(X1, X2) 0.00/0.74 , first(0(), Z) -> nil() 0.00/0.74 , s(X) -> n__s(X) 0.00/0.74 , activate(X) -> X 0.00/0.74 , activate(n__from(X)) -> from(activate(X)) 0.00/0.74 , activate(n__s(X)) -> s(activate(X)) 0.00/0.74 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) } 0.00/0.74 0.00/0.74 We are left with following problem, upon which TcT provides the 0.00/0.74 certificate YES(O(1),O(n^1)). 0.00/0.74 0.00/0.74 Strict DPs: 0.00/0.74 { from^#(X) -> c_1() 0.00/0.74 , from^#(X) -> c_2() 0.00/0.74 , first^#(X1, X2) -> c_3() 0.00/0.74 , first^#(0(), Z) -> c_4() 0.00/0.74 , s^#(X) -> c_5() 0.00/0.74 , activate^#(X) -> c_6() 0.00/0.74 , activate^#(n__from(X)) -> c_7(from^#(activate(X))) 0.00/0.74 , activate^#(n__s(X)) -> c_8(s^#(activate(X))) 0.00/0.74 , activate^#(n__first(X1, X2)) -> 0.00/0.74 c_9(first^#(activate(X1), activate(X2))) 0.00/0.74 , sel^#(0(), cons(X, Z)) -> c_10() } 0.00/0.74 Strict Trs: 0.00/0.74 { from(X) -> cons(X, n__from(n__s(X))) 0.00/0.74 , from(X) -> n__from(X) 0.00/0.74 , first(X1, X2) -> n__first(X1, X2) 0.00/0.74 , first(0(), Z) -> nil() 0.00/0.74 , s(X) -> n__s(X) 0.00/0.74 , activate(X) -> X 0.00/0.74 , activate(n__from(X)) -> from(activate(X)) 0.00/0.74 , activate(n__s(X)) -> s(activate(X)) 0.00/0.74 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) } 0.00/0.74 Obligation: 0.00/0.74 innermost runtime complexity 0.00/0.74 Answer: 0.00/0.74 YES(O(1),O(n^1)) 0.00/0.74 0.00/0.74 The weightgap principle applies (using the following constant 0.00/0.74 growth matrix-interpretation) 0.00/0.74 0.00/0.74 The following argument positions are usable: 0.00/0.74 Uargs(from) = {1}, Uargs(first) = {1, 2}, Uargs(s) = {1}, 0.00/0.74 Uargs(from^#) = {1}, Uargs(first^#) = {1, 2}, Uargs(s^#) = {1}, 0.00/0.74 Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {1} 0.00/0.74 0.00/0.74 TcT has computed the following constructor-restricted matrix 0.00/0.74 interpretation. 0.00/0.74 0.00/0.74 [from](x1) = [1 0] x1 + [2] 0.00/0.74 [0 1] [2] 0.00/0.74 0.00/0.74 [cons](x1, x2) = [1 0] x1 + [1] 0.00/0.74 [0 1] [1] 0.00/0.74 0.00/0.74 [n__from](x1) = [1 0] x1 + [0] 0.00/0.74 [0 1] [2] 0.00/0.74 0.00/0.74 [n__s](x1) = [1 0] x1 + [0] 0.00/0.74 [0 1] [2] 0.00/0.74 0.00/0.74 [first](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 0.00/0.74 [0 1] [0 1] [2] 0.00/0.74 0.00/0.74 [0] = [0] 0.00/0.74 [0] 0.00/0.74 0.00/0.74 [nil] = [1] 0.00/0.74 [1] 0.00/0.74 0.00/0.74 [s](x1) = [1 0] x1 + [1] 0.00/0.74 [0 1] [2] 0.00/0.74 0.00/0.74 [n__first](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 0.00/0.74 [0 1] [0 1] [2] 0.00/0.74 0.00/0.74 [activate](x1) = [1 2] x1 + [1] 0.00/0.74 [0 2] [0] 0.00/0.74 0.00/0.74 [from^#](x1) = [1 0] x1 + [1] 0.00/0.74 [0 0] [2] 0.00/0.74 0.00/0.74 [c_1] = [1] 0.00/0.74 [1] 0.00/0.74 0.00/0.74 [c_2] = [1] 0.00/0.74 [1] 0.00/0.74 0.00/0.74 [first^#](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 0.00/0.74 [0 0] [0 0] [2] 0.00/0.74 0.00/0.74 [c_3] = [1] 0.00/0.74 [1] 0.00/0.74 0.00/0.74 [c_4] = [1] 0.00/0.74 [1] 0.00/0.74 0.00/0.74 [s^#](x1) = [1 0] x1 + [1] 0.00/0.74 [0 0] [2] 0.00/0.74 0.00/0.74 [c_5] = [1] 0.00/0.74 [1] 0.00/0.74 0.00/0.74 [activate^#](x1) = [1 2] x1 + [0] 0.00/0.74 [0 0] [0] 0.00/0.74 0.00/0.74 [c_6] = [1] 0.00/0.74 [1] 0.00/0.74 0.00/0.74 [c_7](x1) = [1 0] x1 + [2] 0.00/0.74 [0 1] [2] 0.00/0.74 0.00/0.74 [c_8](x1) = [1 0] x1 + [2] 0.00/0.74 [0 1] [2] 0.00/0.74 0.00/0.74 [c_9](x1) = [1 0] x1 + [2] 0.00/0.74 [0 1] [2] 0.00/0.74 0.00/0.74 [sel^#](x1, x2) = [2 2] x1 + [2 2] x2 + [2] 0.00/0.74 [2 2] [2 2] [2] 0.00/0.74 0.00/0.74 [c_10] = [1] 0.00/0.75 [1] 0.00/0.75 0.00/0.75 The order satisfies the following ordering constraints: 0.00/0.75 0.00/0.75 [from(X)] = [1 0] X + [2] 0.00/0.75 [0 1] [2] 0.00/0.75 > [1 0] X + [1] 0.00/0.75 [0 1] [1] 0.00/0.75 = [cons(X, n__from(n__s(X)))] 0.00/0.75 0.00/0.75 [from(X)] = [1 0] X + [2] 0.00/0.75 [0 1] [2] 0.00/0.75 > [1 0] X + [0] 0.00/0.75 [0 1] [2] 0.00/0.75 = [n__from(X)] 0.00/0.75 0.00/0.75 [first(X1, X2)] = [1 0] X1 + [1 0] X2 + [2] 0.00/0.75 [0 1] [0 1] [2] 0.00/0.75 > [1 0] X1 + [1 0] X2 + [0] 0.00/0.75 [0 1] [0 1] [2] 0.00/0.75 = [n__first(X1, X2)] 0.00/0.75 0.00/0.75 [first(0(), Z)] = [1 0] Z + [2] 0.00/0.75 [0 1] [2] 0.00/0.75 > [1] 0.00/0.75 [1] 0.00/0.75 = [nil()] 0.00/0.75 0.00/0.75 [s(X)] = [1 0] X + [1] 0.00/0.75 [0 1] [2] 0.00/0.75 > [1 0] X + [0] 0.00/0.75 [0 1] [2] 0.00/0.75 = [n__s(X)] 0.00/0.75 0.00/0.75 [activate(X)] = [1 2] X + [1] 0.00/0.75 [0 2] [0] 0.00/0.75 > [1 0] X + [0] 0.00/0.75 [0 1] [0] 0.00/0.75 = [X] 0.00/0.75 0.00/0.75 [activate(n__from(X))] = [1 2] X + [5] 0.00/0.75 [0 2] [4] 0.00/0.75 > [1 2] X + [3] 0.00/0.75 [0 2] [2] 0.00/0.75 = [from(activate(X))] 0.00/0.75 0.00/0.75 [activate(n__s(X))] = [1 2] X + [5] 0.00/0.75 [0 2] [4] 0.00/0.75 > [1 2] X + [2] 0.00/0.75 [0 2] [2] 0.00/0.75 = [s(activate(X))] 0.00/0.75 0.00/0.75 [activate(n__first(X1, X2))] = [1 2] X1 + [1 2] X2 + [5] 0.00/0.75 [0 2] [0 2] [4] 0.00/0.75 > [1 2] X1 + [1 2] X2 + [4] 0.00/0.75 [0 2] [0 2] [2] 0.00/0.75 = [first(activate(X1), activate(X2))] 0.00/0.75 0.00/0.75 [from^#(X)] = [1 0] X + [1] 0.00/0.75 [0 0] [2] 0.00/0.75 >= [1] 0.00/0.75 [1] 0.00/0.75 = [c_1()] 0.00/0.75 0.00/0.75 [from^#(X)] = [1 0] X + [1] 0.00/0.75 [0 0] [2] 0.00/0.75 >= [1] 0.00/0.75 [1] 0.00/0.75 = [c_2()] 0.00/0.75 0.00/0.75 [first^#(X1, X2)] = [1 0] X1 + [1 0] X2 + [2] 0.00/0.75 [0 0] [0 0] [2] 0.00/0.75 > [1] 0.00/0.75 [1] 0.00/0.75 = [c_3()] 0.00/0.75 0.00/0.75 [first^#(0(), Z)] = [1 0] Z + [2] 0.00/0.75 [0 0] [2] 0.00/0.75 > [1] 0.00/0.75 [1] 0.00/0.75 = [c_4()] 0.00/0.75 0.00/0.75 [s^#(X)] = [1 0] X + [1] 0.00/0.75 [0 0] [2] 0.00/0.75 >= [1] 0.00/0.75 [1] 0.00/0.75 = [c_5()] 0.00/0.75 0.00/0.75 [activate^#(X)] = [1 2] X + [0] 0.00/0.75 [0 0] [0] 0.00/0.75 ? [1] 0.00/0.75 [1] 0.00/0.75 = [c_6()] 0.00/0.75 0.00/0.75 [activate^#(n__from(X))] = [1 2] X + [4] 0.00/0.75 [0 0] [0] 0.00/0.75 ? [1 2] X + [4] 0.00/0.75 [0 0] [4] 0.00/0.75 = [c_7(from^#(activate(X)))] 0.00/0.75 0.00/0.75 [activate^#(n__s(X))] = [1 2] X + [4] 0.00/0.75 [0 0] [0] 0.00/0.75 ? [1 2] X + [4] 0.00/0.75 [0 0] [4] 0.00/0.75 = [c_8(s^#(activate(X)))] 0.00/0.75 0.00/0.75 [activate^#(n__first(X1, X2))] = [1 2] X1 + [1 2] X2 + [4] 0.00/0.75 [0 0] [0 0] [0] 0.00/0.75 ? [1 2] X1 + [1 2] X2 + [6] 0.00/0.75 [0 0] [0 0] [4] 0.00/0.75 = [c_9(first^#(activate(X1), activate(X2)))] 0.00/0.75 0.00/0.75 [sel^#(0(), cons(X, Z))] = [2 2] X + [6] 0.00/0.75 [2 2] [6] 0.00/0.75 > [1] 0.00/0.75 [1] 0.00/0.75 = [c_10()] 0.00/0.75 0.00/0.75 0.00/0.75 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.75 0.00/0.75 We are left with following problem, upon which TcT provides the 0.00/0.75 certificate YES(O(1),O(1)). 0.00/0.75 0.00/0.75 Strict DPs: 0.00/0.75 { from^#(X) -> c_1() 0.00/0.75 , from^#(X) -> c_2() 0.00/0.75 , s^#(X) -> c_5() 0.00/0.75 , activate^#(X) -> c_6() 0.00/0.75 , activate^#(n__from(X)) -> c_7(from^#(activate(X))) 0.00/0.75 , activate^#(n__s(X)) -> c_8(s^#(activate(X))) 0.00/0.75 , activate^#(n__first(X1, X2)) -> 0.00/0.75 c_9(first^#(activate(X1), activate(X2))) } 0.00/0.75 Weak DPs: 0.00/0.75 { first^#(X1, X2) -> c_3() 0.00/0.75 , first^#(0(), Z) -> c_4() 0.00/0.75 , sel^#(0(), cons(X, Z)) -> c_10() } 0.00/0.75 Weak Trs: 0.00/0.75 { from(X) -> cons(X, n__from(n__s(X))) 0.00/0.75 , from(X) -> n__from(X) 0.00/0.75 , first(X1, X2) -> n__first(X1, X2) 0.00/0.75 , first(0(), Z) -> nil() 0.00/0.75 , s(X) -> n__s(X) 0.00/0.75 , activate(X) -> X 0.00/0.75 , activate(n__from(X)) -> from(activate(X)) 0.00/0.75 , activate(n__s(X)) -> s(activate(X)) 0.00/0.75 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) } 0.00/0.75 Obligation: 0.00/0.75 innermost runtime complexity 0.00/0.75 Answer: 0.00/0.75 YES(O(1),O(1)) 0.00/0.75 0.00/0.75 We estimate the number of application of {1,2,3,4,7} by 0.00/0.75 applications of Pre({1,2,3,4,7}) = {5,6}. Here rules are labeled as 0.00/0.75 follows: 0.00/0.75 0.00/0.75 DPs: 0.00/0.75 { 1: from^#(X) -> c_1() 0.00/0.75 , 2: from^#(X) -> c_2() 0.00/0.75 , 3: s^#(X) -> c_5() 0.00/0.75 , 4: activate^#(X) -> c_6() 0.00/0.75 , 5: activate^#(n__from(X)) -> c_7(from^#(activate(X))) 0.00/0.75 , 6: activate^#(n__s(X)) -> c_8(s^#(activate(X))) 0.00/0.75 , 7: activate^#(n__first(X1, X2)) -> 0.00/0.75 c_9(first^#(activate(X1), activate(X2))) 0.00/0.75 , 8: first^#(X1, X2) -> c_3() 0.00/0.75 , 9: first^#(0(), Z) -> c_4() 0.00/0.75 , 10: sel^#(0(), cons(X, Z)) -> c_10() } 0.00/0.75 0.00/0.75 We are left with following problem, upon which TcT provides the 0.00/0.75 certificate YES(O(1),O(1)). 0.00/0.75 0.00/0.75 Strict DPs: 0.00/0.75 { activate^#(n__from(X)) -> c_7(from^#(activate(X))) 0.00/0.75 , activate^#(n__s(X)) -> c_8(s^#(activate(X))) } 0.00/0.75 Weak DPs: 0.00/0.75 { from^#(X) -> c_1() 0.00/0.75 , from^#(X) -> c_2() 0.00/0.75 , first^#(X1, X2) -> c_3() 0.00/0.75 , first^#(0(), Z) -> c_4() 0.00/0.75 , s^#(X) -> c_5() 0.00/0.75 , activate^#(X) -> c_6() 0.00/0.75 , activate^#(n__first(X1, X2)) -> 0.00/0.75 c_9(first^#(activate(X1), activate(X2))) 0.00/0.75 , sel^#(0(), cons(X, Z)) -> c_10() } 0.00/0.75 Weak Trs: 0.00/0.75 { from(X) -> cons(X, n__from(n__s(X))) 0.00/0.75 , from(X) -> n__from(X) 0.00/0.75 , first(X1, X2) -> n__first(X1, X2) 0.00/0.75 , first(0(), Z) -> nil() 0.00/0.75 , s(X) -> n__s(X) 0.00/0.75 , activate(X) -> X 0.00/0.75 , activate(n__from(X)) -> from(activate(X)) 0.00/0.75 , activate(n__s(X)) -> s(activate(X)) 0.00/0.75 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) } 0.00/0.75 Obligation: 0.00/0.75 innermost runtime complexity 0.00/0.75 Answer: 0.00/0.75 YES(O(1),O(1)) 0.00/0.75 0.00/0.75 We estimate the number of application of {1,2} by applications of 0.00/0.75 Pre({1,2}) = {}. Here rules are labeled as follows: 0.00/0.75 0.00/0.75 DPs: 0.00/0.75 { 1: activate^#(n__from(X)) -> c_7(from^#(activate(X))) 0.00/0.75 , 2: activate^#(n__s(X)) -> c_8(s^#(activate(X))) 0.00/0.75 , 3: from^#(X) -> c_1() 0.00/0.75 , 4: from^#(X) -> c_2() 0.00/0.75 , 5: first^#(X1, X2) -> c_3() 0.00/0.75 , 6: first^#(0(), Z) -> c_4() 0.00/0.75 , 7: s^#(X) -> c_5() 0.00/0.75 , 8: activate^#(X) -> c_6() 0.00/0.75 , 9: activate^#(n__first(X1, X2)) -> 0.00/0.75 c_9(first^#(activate(X1), activate(X2))) 0.00/0.75 , 10: sel^#(0(), cons(X, Z)) -> c_10() } 0.00/0.75 0.00/0.75 We are left with following problem, upon which TcT provides the 0.00/0.75 certificate YES(O(1),O(1)). 0.00/0.75 0.00/0.75 Weak DPs: 0.00/0.75 { from^#(X) -> c_1() 0.00/0.75 , from^#(X) -> c_2() 0.00/0.75 , first^#(X1, X2) -> c_3() 0.00/0.75 , first^#(0(), Z) -> c_4() 0.00/0.75 , s^#(X) -> c_5() 0.00/0.75 , activate^#(X) -> c_6() 0.00/0.75 , activate^#(n__from(X)) -> c_7(from^#(activate(X))) 0.00/0.75 , activate^#(n__s(X)) -> c_8(s^#(activate(X))) 0.00/0.75 , activate^#(n__first(X1, X2)) -> 0.00/0.75 c_9(first^#(activate(X1), activate(X2))) 0.00/0.75 , sel^#(0(), cons(X, Z)) -> c_10() } 0.00/0.75 Weak Trs: 0.00/0.75 { from(X) -> cons(X, n__from(n__s(X))) 0.00/0.75 , from(X) -> n__from(X) 0.00/0.75 , first(X1, X2) -> n__first(X1, X2) 0.00/0.75 , first(0(), Z) -> nil() 0.00/0.75 , s(X) -> n__s(X) 0.00/0.75 , activate(X) -> X 0.00/0.75 , activate(n__from(X)) -> from(activate(X)) 0.00/0.75 , activate(n__s(X)) -> s(activate(X)) 0.00/0.75 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) } 0.00/0.75 Obligation: 0.00/0.75 innermost runtime complexity 0.00/0.75 Answer: 0.00/0.75 YES(O(1),O(1)) 0.00/0.75 0.00/0.75 The following weak DPs constitute a sub-graph of the DG that is 0.00/0.75 closed under successors. The DPs are removed. 0.00/0.75 0.00/0.75 { from^#(X) -> c_1() 0.00/0.75 , from^#(X) -> c_2() 0.00/0.75 , first^#(X1, X2) -> c_3() 0.00/0.75 , first^#(0(), Z) -> c_4() 0.00/0.75 , s^#(X) -> c_5() 0.00/0.75 , activate^#(X) -> c_6() 0.00/0.75 , activate^#(n__from(X)) -> c_7(from^#(activate(X))) 0.00/0.75 , activate^#(n__s(X)) -> c_8(s^#(activate(X))) 0.00/0.75 , activate^#(n__first(X1, X2)) -> 0.00/0.75 c_9(first^#(activate(X1), activate(X2))) 0.00/0.75 , sel^#(0(), cons(X, Z)) -> c_10() } 0.00/0.75 0.00/0.75 We are left with following problem, upon which TcT provides the 0.00/0.75 certificate YES(O(1),O(1)). 0.00/0.75 0.00/0.75 Weak Trs: 0.00/0.75 { from(X) -> cons(X, n__from(n__s(X))) 0.00/0.75 , from(X) -> n__from(X) 0.00/0.75 , first(X1, X2) -> n__first(X1, X2) 0.00/0.75 , first(0(), Z) -> nil() 0.00/0.75 , s(X) -> n__s(X) 0.00/0.75 , activate(X) -> X 0.00/0.75 , activate(n__from(X)) -> from(activate(X)) 0.00/0.75 , activate(n__s(X)) -> s(activate(X)) 0.00/0.75 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) } 0.00/0.75 Obligation: 0.00/0.75 innermost runtime complexity 0.00/0.75 Answer: 0.00/0.75 YES(O(1),O(1)) 0.00/0.75 0.00/0.75 No rule is usable, rules are removed from the input problem. 0.00/0.75 0.00/0.75 We are left with following problem, upon which TcT provides the 0.00/0.75 certificate YES(O(1),O(1)). 0.00/0.75 0.00/0.75 Rules: Empty 0.00/0.75 Obligation: 0.00/0.75 innermost runtime complexity 0.00/0.75 Answer: 0.00/0.75 YES(O(1),O(1)) 0.00/0.75 0.00/0.75 Empty rules are trivially bounded 0.00/0.75 0.00/0.75 Hurray, we answered YES(O(1),O(n^1)) 0.00/0.75 EOF