YES(O(1),O(n^2)) 220.12/148.07 YES(O(1),O(n^2)) 220.12/148.07 220.12/148.07 We are left with following problem, upon which TcT provides the 220.12/148.07 certificate YES(O(1),O(n^2)). 220.12/148.07 220.12/148.07 Strict Trs: 220.12/148.07 { from(X) -> cons(X, n__from(s(X))) 220.12/148.07 , from(X) -> n__from(X) 220.12/148.07 , first(X1, X2) -> n__first(X1, X2) 220.12/148.07 , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) 220.12/148.07 , first(0(), Z) -> nil() 220.12/148.07 , activate(X) -> X 220.12/148.07 , activate(n__from(X)) -> from(X) 220.12/148.07 , activate(n__first(X1, X2)) -> first(X1, X2) 220.12/148.07 , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) 220.12/148.07 , sel(0(), cons(X, Z)) -> X } 220.12/148.07 Obligation: 220.12/148.07 innermost runtime complexity 220.12/148.07 Answer: 220.12/148.07 YES(O(1),O(n^2)) 220.12/148.07 220.12/148.07 We add the following weak dependency pairs: 220.12/148.07 220.12/148.07 Strict DPs: 220.12/148.07 { from^#(X) -> c_1() 220.12/148.07 , from^#(X) -> c_2() 220.12/148.07 , first^#(X1, X2) -> c_3() 220.12/148.07 , first^#(s(X), cons(Y, Z)) -> c_4(activate^#(Z)) 220.12/148.07 , first^#(0(), Z) -> c_5() 220.12/148.07 , activate^#(X) -> c_6() 220.12/148.07 , activate^#(n__from(X)) -> c_7(from^#(X)) 220.12/148.07 , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2)) 220.12/148.07 , sel^#(s(X), cons(Y, Z)) -> c_9(sel^#(X, activate(Z))) 220.12/148.07 , sel^#(0(), cons(X, Z)) -> c_10() } 220.12/148.07 220.12/148.07 and mark the set of starting terms. 220.12/148.07 220.12/148.07 We are left with following problem, upon which TcT provides the 220.12/148.07 certificate YES(O(1),O(n^2)). 220.12/148.07 220.12/148.07 Strict DPs: 220.12/148.07 { from^#(X) -> c_1() 220.12/148.07 , from^#(X) -> c_2() 220.12/148.07 , first^#(X1, X2) -> c_3() 220.12/148.07 , first^#(s(X), cons(Y, Z)) -> c_4(activate^#(Z)) 220.12/148.07 , first^#(0(), Z) -> c_5() 220.12/148.07 , activate^#(X) -> c_6() 220.12/148.07 , activate^#(n__from(X)) -> c_7(from^#(X)) 220.12/148.07 , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2)) 220.12/148.07 , sel^#(s(X), cons(Y, Z)) -> c_9(sel^#(X, activate(Z))) 220.12/148.07 , sel^#(0(), cons(X, Z)) -> c_10() } 220.12/148.07 Strict Trs: 220.12/148.07 { from(X) -> cons(X, n__from(s(X))) 220.12/148.07 , from(X) -> n__from(X) 220.12/148.07 , first(X1, X2) -> n__first(X1, X2) 220.12/148.07 , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) 220.12/148.07 , first(0(), Z) -> nil() 220.12/148.07 , activate(X) -> X 220.12/148.07 , activate(n__from(X)) -> from(X) 220.12/148.07 , activate(n__first(X1, X2)) -> first(X1, X2) 220.12/148.07 , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) 220.12/148.07 , sel(0(), cons(X, Z)) -> X } 220.12/148.07 Obligation: 220.12/148.07 innermost runtime complexity 220.12/148.07 Answer: 220.12/148.07 YES(O(1),O(n^2)) 220.12/148.07 220.12/148.07 We replace rewrite rules by usable rules: 220.12/148.07 220.12/148.07 Strict Usable Rules: 220.12/148.07 { from(X) -> cons(X, n__from(s(X))) 220.12/148.07 , from(X) -> n__from(X) 220.12/148.07 , first(X1, X2) -> n__first(X1, X2) 220.12/148.07 , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) 220.12/148.07 , first(0(), Z) -> nil() 220.12/148.07 , activate(X) -> X 220.12/148.07 , activate(n__from(X)) -> from(X) 220.12/148.07 , activate(n__first(X1, X2)) -> first(X1, X2) } 220.12/148.07 220.12/148.07 We are left with following problem, upon which TcT provides the 220.12/148.07 certificate YES(O(1),O(n^2)). 220.12/148.07 220.12/148.07 Strict DPs: 220.12/148.07 { from^#(X) -> c_1() 220.12/148.07 , from^#(X) -> c_2() 220.12/148.07 , first^#(X1, X2) -> c_3() 220.12/148.07 , first^#(s(X), cons(Y, Z)) -> c_4(activate^#(Z)) 220.12/148.07 , first^#(0(), Z) -> c_5() 220.12/148.07 , activate^#(X) -> c_6() 220.12/148.07 , activate^#(n__from(X)) -> c_7(from^#(X)) 220.12/148.07 , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2)) 220.12/148.07 , sel^#(s(X), cons(Y, Z)) -> c_9(sel^#(X, activate(Z))) 220.12/148.07 , sel^#(0(), cons(X, Z)) -> c_10() } 220.12/148.07 Strict Trs: 220.12/148.07 { from(X) -> cons(X, n__from(s(X))) 220.12/148.07 , from(X) -> n__from(X) 220.12/148.07 , first(X1, X2) -> n__first(X1, X2) 220.12/148.07 , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) 220.12/148.07 , first(0(), Z) -> nil() 220.12/148.07 , activate(X) -> X 220.12/148.07 , activate(n__from(X)) -> from(X) 220.12/148.07 , activate(n__first(X1, X2)) -> first(X1, X2) } 220.12/148.07 Obligation: 220.12/148.07 innermost runtime complexity 220.12/148.07 Answer: 220.12/148.07 YES(O(1),O(n^2)) 220.12/148.07 220.12/148.07 The weightgap principle applies (using the following constant 220.12/148.07 growth matrix-interpretation) 220.12/148.07 220.12/148.07 The following argument positions are usable: 220.12/148.07 Uargs(cons) = {2}, Uargs(n__first) = {2}, Uargs(c_4) = {1}, 220.12/148.07 Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(sel^#) = {2}, 220.12/148.07 Uargs(c_9) = {1} 220.12/148.07 220.12/148.07 TcT has computed the following constructor-restricted matrix 220.12/148.07 interpretation. 220.12/148.07 220.12/148.07 [from](x1) = [1] 220.12/148.07 [2] 220.12/148.07 220.12/148.07 [cons](x1, x2) = [1 0] x2 + [0] 220.12/148.07 [0 0] [2] 220.12/148.07 220.12/148.07 [n__from](x1) = [0] 220.12/148.07 [0] 220.12/148.07 220.12/148.07 [s](x1) = [1 2] x1 + [0] 220.12/148.07 [0 1] [2] 220.12/148.07 220.12/148.07 [first](x1, x2) = [1 2] x1 + [1 0] x2 + [1] 220.12/148.07 [0 0] [0 2] [0] 220.12/148.07 220.12/148.07 [0] = [0] 220.12/148.07 [0] 220.12/148.07 220.12/148.07 [nil] = [0] 220.12/148.07 [0] 220.12/148.07 220.12/148.07 [n__first](x1, x2) = [1 2] x1 + [1 0] x2 + [0] 220.12/148.07 [0 0] [0 1] [0] 220.12/148.07 220.12/148.07 [activate](x1) = [1 0] x1 + [2] 220.12/148.07 [0 2] [2] 220.12/148.07 220.12/148.07 [from^#](x1) = [0 0] x1 + [1] 220.12/148.07 [1 1] [1] 220.12/148.07 220.12/148.07 [c_1] = [0] 220.12/148.07 [1] 220.12/148.07 220.12/148.07 [c_2] = [0] 220.12/148.07 [1] 220.12/148.07 220.12/148.07 [first^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] 220.12/148.07 [1 1] [2 1] [2] 220.12/148.07 220.12/148.07 [c_3] = [1] 220.12/148.07 [1] 220.12/148.07 220.12/148.07 [c_4](x1) = [1 0] x1 + [1] 220.12/148.07 [0 1] [1] 220.12/148.07 220.12/148.07 [activate^#](x1) = [0 0] x1 + [2] 220.12/148.07 [2 1] [2] 220.12/148.07 220.12/148.07 [c_5] = [1] 220.12/148.07 [1] 220.12/148.07 220.12/148.07 [c_6] = [2] 220.12/148.07 [1] 220.12/148.07 220.12/148.07 [c_7](x1) = [1 0] x1 + [2] 220.12/148.07 [0 1] [2] 220.12/148.07 220.12/148.07 [c_8](x1) = [1 0] x1 + [1] 220.12/148.07 [0 1] [1] 220.12/148.07 220.12/148.07 [sel^#](x1, x2) = [2 0] x2 + [0] 220.12/148.07 [0 0] [0] 220.12/148.07 220.12/148.07 [c_9](x1) = [1 0] x1 + [2] 220.12/148.07 [0 1] [2] 220.12/148.07 220.12/148.07 [c_10] = [2] 220.12/148.07 [0] 220.12/148.07 220.12/148.07 The order satisfies the following ordering constraints: 220.12/148.07 220.12/148.07 [from(X)] = [1] 220.12/148.07 [2] 220.12/148.07 > [0] 220.12/148.07 [2] 220.12/148.07 = [cons(X, n__from(s(X)))] 220.12/148.07 220.12/148.07 [from(X)] = [1] 220.12/148.07 [2] 220.12/148.07 > [0] 220.12/148.07 [0] 220.12/148.07 = [n__from(X)] 220.12/148.07 220.12/148.07 [first(X1, X2)] = [1 2] X1 + [1 0] X2 + [1] 220.12/148.07 [0 0] [0 2] [0] 220.12/148.07 > [1 2] X1 + [1 0] X2 + [0] 220.12/148.07 [0 0] [0 1] [0] 220.12/148.07 = [n__first(X1, X2)] 220.12/148.07 220.12/148.07 [first(s(X), cons(Y, Z))] = [1 4] X + [1 0] Z + [5] 220.12/148.07 [0 0] [0 0] [4] 220.12/148.07 > [1 2] X + [1 0] Z + [2] 220.12/148.07 [0 0] [0 0] [2] 220.12/148.07 = [cons(Y, n__first(X, activate(Z)))] 220.12/148.07 220.12/148.07 [first(0(), Z)] = [1 0] Z + [1] 220.12/148.07 [0 2] [0] 220.12/148.07 > [0] 220.12/148.07 [0] 220.12/148.07 = [nil()] 220.12/148.07 220.12/148.07 [activate(X)] = [1 0] X + [2] 220.12/148.07 [0 2] [2] 220.12/148.07 > [1 0] X + [0] 220.12/148.07 [0 1] [0] 220.12/148.07 = [X] 220.12/148.07 220.12/148.07 [activate(n__from(X))] = [2] 220.12/148.07 [2] 220.12/148.07 > [1] 220.12/148.07 [2] 220.12/148.07 = [from(X)] 220.12/148.07 220.12/148.07 [activate(n__first(X1, X2))] = [1 2] X1 + [1 0] X2 + [2] 220.12/148.07 [0 0] [0 2] [2] 220.12/148.07 > [1 2] X1 + [1 0] X2 + [1] 220.12/148.07 [0 0] [0 2] [0] 220.12/148.07 = [first(X1, X2)] 220.12/148.07 220.12/148.07 [from^#(X)] = [0 0] X + [1] 220.12/148.07 [1 1] [1] 220.12/148.07 > [0] 220.12/148.07 [1] 220.12/148.07 = [c_1()] 220.12/148.07 220.12/148.07 [from^#(X)] = [0 0] X + [1] 220.12/148.07 [1 1] [1] 220.12/148.07 > [0] 220.12/148.07 [1] 220.12/148.07 = [c_2()] 220.12/148.07 220.12/148.07 [first^#(X1, X2)] = [0 0] X1 + [0 0] X2 + [2] 220.12/148.07 [1 1] [2 1] [2] 220.12/148.07 > [1] 220.12/148.07 [1] 220.12/148.07 = [c_3()] 220.12/148.07 220.12/148.07 [first^#(s(X), cons(Y, Z))] = [0 0] X + [0 0] Z + [2] 220.12/148.07 [1 3] [2 0] [6] 220.12/148.07 ? [0 0] Z + [3] 220.12/148.07 [2 1] [3] 220.12/148.07 = [c_4(activate^#(Z))] 220.12/148.07 220.12/148.07 [first^#(0(), Z)] = [0 0] Z + [2] 220.12/148.07 [2 1] [2] 220.12/148.07 > [1] 220.12/148.07 [1] 220.12/148.07 = [c_5()] 220.12/148.07 220.12/148.07 [activate^#(X)] = [0 0] X + [2] 220.12/148.07 [2 1] [2] 220.12/148.07 >= [2] 220.12/148.07 [1] 220.12/148.07 = [c_6()] 220.12/148.07 220.12/148.07 [activate^#(n__from(X))] = [2] 220.12/148.07 [2] 220.12/148.07 ? [0 0] X + [3] 220.12/148.07 [1 1] [3] 220.12/148.07 = [c_7(from^#(X))] 220.12/148.07 220.12/148.07 [activate^#(n__first(X1, X2))] = [0 0] X1 + [0 0] X2 + [2] 220.12/148.07 [2 4] [2 1] [2] 220.12/148.07 ? [0 0] X1 + [0 0] X2 + [3] 220.12/148.07 [1 1] [2 1] [3] 220.12/148.07 = [c_8(first^#(X1, X2))] 220.12/148.07 220.12/148.07 [sel^#(s(X), cons(Y, Z))] = [2 0] Z + [0] 220.12/148.07 [0 0] [0] 220.12/148.07 ? [2 0] Z + [6] 220.12/148.07 [0 0] [2] 220.12/148.07 = [c_9(sel^#(X, activate(Z)))] 220.12/148.07 220.12/148.07 [sel^#(0(), cons(X, Z))] = [2 0] Z + [0] 220.12/148.07 [0 0] [0] 220.12/148.07 ? [2] 220.12/148.07 [0] 220.12/148.07 = [c_10()] 220.12/148.07 220.12/148.07 220.12/148.07 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 220.12/148.07 220.12/148.07 We are left with following problem, upon which TcT provides the 220.12/148.07 certificate YES(O(1),O(n^1)). 220.12/148.07 220.12/148.07 Strict DPs: 220.12/148.07 { first^#(s(X), cons(Y, Z)) -> c_4(activate^#(Z)) 220.12/148.07 , activate^#(X) -> c_6() 220.12/148.07 , activate^#(n__from(X)) -> c_7(from^#(X)) 220.12/148.07 , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2)) 220.12/148.07 , sel^#(s(X), cons(Y, Z)) -> c_9(sel^#(X, activate(Z))) 220.12/148.07 , sel^#(0(), cons(X, Z)) -> c_10() } 220.12/148.07 Weak DPs: 220.12/148.07 { from^#(X) -> c_1() 220.12/148.07 , from^#(X) -> c_2() 220.12/148.07 , first^#(X1, X2) -> c_3() 220.12/148.07 , first^#(0(), Z) -> c_5() } 220.12/148.07 Weak Trs: 220.12/148.07 { from(X) -> cons(X, n__from(s(X))) 220.12/148.07 , from(X) -> n__from(X) 220.12/148.07 , first(X1, X2) -> n__first(X1, X2) 220.12/148.07 , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) 220.12/148.07 , first(0(), Z) -> nil() 220.12/148.07 , activate(X) -> X 220.12/148.07 , activate(n__from(X)) -> from(X) 220.12/148.07 , activate(n__first(X1, X2)) -> first(X1, X2) } 220.12/148.07 Obligation: 220.12/148.07 innermost runtime complexity 220.12/148.07 Answer: 220.12/148.07 YES(O(1),O(n^1)) 220.12/148.07 220.12/148.07 We estimate the number of application of {2,3,6} by applications of 220.12/148.07 Pre({2,3,6}) = {1,5}. Here rules are labeled as follows: 220.12/148.07 220.12/148.07 DPs: 220.12/148.07 { 1: first^#(s(X), cons(Y, Z)) -> c_4(activate^#(Z)) 220.12/148.07 , 2: activate^#(X) -> c_6() 220.12/148.07 , 3: activate^#(n__from(X)) -> c_7(from^#(X)) 220.12/148.07 , 4: activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2)) 220.12/148.07 , 5: sel^#(s(X), cons(Y, Z)) -> c_9(sel^#(X, activate(Z))) 220.12/148.07 , 6: sel^#(0(), cons(X, Z)) -> c_10() 220.12/148.07 , 7: from^#(X) -> c_1() 220.12/148.07 , 8: from^#(X) -> c_2() 220.12/148.07 , 9: first^#(X1, X2) -> c_3() 220.12/148.07 , 10: first^#(0(), Z) -> c_5() } 220.12/148.07 220.12/148.07 We are left with following problem, upon which TcT provides the 220.12/148.07 certificate YES(O(1),O(n^1)). 220.12/148.07 220.12/148.07 Strict DPs: 220.12/148.07 { first^#(s(X), cons(Y, Z)) -> c_4(activate^#(Z)) 220.12/148.07 , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2)) 220.12/148.07 , sel^#(s(X), cons(Y, Z)) -> c_9(sel^#(X, activate(Z))) } 220.12/148.07 Weak DPs: 220.12/148.07 { from^#(X) -> c_1() 220.12/148.07 , from^#(X) -> c_2() 220.12/148.07 , first^#(X1, X2) -> c_3() 220.12/148.07 , first^#(0(), Z) -> c_5() 220.12/148.07 , activate^#(X) -> c_6() 220.12/148.07 , activate^#(n__from(X)) -> c_7(from^#(X)) 220.12/148.07 , sel^#(0(), cons(X, Z)) -> c_10() } 220.12/148.07 Weak Trs: 220.12/148.07 { from(X) -> cons(X, n__from(s(X))) 220.12/148.07 , from(X) -> n__from(X) 220.12/148.07 , first(X1, X2) -> n__first(X1, X2) 220.12/148.07 , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) 220.12/148.07 , first(0(), Z) -> nil() 220.12/148.07 , activate(X) -> X 220.12/148.07 , activate(n__from(X)) -> from(X) 220.12/148.07 , activate(n__first(X1, X2)) -> first(X1, X2) } 220.12/148.07 Obligation: 220.12/148.07 innermost runtime complexity 220.12/148.07 Answer: 220.12/148.07 YES(O(1),O(n^1)) 220.12/148.07 220.12/148.07 The following weak DPs constitute a sub-graph of the DG that is 220.12/148.07 closed under successors. The DPs are removed. 220.12/148.07 220.12/148.07 { from^#(X) -> c_1() 220.12/148.07 , from^#(X) -> c_2() 220.12/148.07 , first^#(X1, X2) -> c_3() 220.12/148.07 , first^#(0(), Z) -> c_5() 220.12/148.07 , activate^#(X) -> c_6() 220.12/148.07 , activate^#(n__from(X)) -> c_7(from^#(X)) 220.12/148.07 , sel^#(0(), cons(X, Z)) -> c_10() } 220.12/148.07 220.12/148.07 We are left with following problem, upon which TcT provides the 220.12/148.07 certificate YES(O(1),O(n^1)). 220.12/148.07 220.12/148.07 Strict DPs: 220.12/148.07 { first^#(s(X), cons(Y, Z)) -> c_4(activate^#(Z)) 220.12/148.07 , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2)) 220.12/148.07 , sel^#(s(X), cons(Y, Z)) -> c_9(sel^#(X, activate(Z))) } 220.12/148.07 Weak Trs: 220.12/148.07 { from(X) -> cons(X, n__from(s(X))) 220.12/148.07 , from(X) -> n__from(X) 220.12/148.07 , first(X1, X2) -> n__first(X1, X2) 220.12/148.07 , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) 220.12/148.07 , first(0(), Z) -> nil() 220.12/148.07 , activate(X) -> X 220.12/148.07 , activate(n__from(X)) -> from(X) 220.12/148.07 , activate(n__first(X1, X2)) -> first(X1, X2) } 220.12/148.07 Obligation: 220.12/148.07 innermost runtime complexity 220.12/148.07 Answer: 220.12/148.07 YES(O(1),O(n^1)) 220.12/148.07 220.12/148.07 We use the processor 'Small Polynomial Path Order (PS,1-bounded)' 220.12/148.07 to orient following rules strictly. 220.12/148.07 220.12/148.07 DPs: 220.12/148.07 { 1: first^#(s(X), cons(Y, Z)) -> c_4(activate^#(Z)) 220.12/148.07 , 2: activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2)) 220.12/148.07 , 3: sel^#(s(X), cons(Y, Z)) -> c_9(sel^#(X, activate(Z))) } 220.12/148.07 Trs: 220.12/148.07 { from(X) -> n__from(X) 220.12/148.07 , first(0(), Z) -> nil() } 220.12/148.07 220.12/148.07 Sub-proof: 220.12/148.07 ---------- 220.12/148.07 The input was oriented with the instance of 'Small Polynomial Path 220.12/148.07 Order (PS,1-bounded)' as induced by the safe mapping 220.12/148.07 220.12/148.07 safe(from) = {1}, safe(cons) = {1, 2}, safe(n__from) = {1}, 220.12/148.07 safe(s) = {1}, safe(first) = {}, safe(0) = {}, safe(nil) = {}, 220.12/148.07 safe(n__first) = {1, 2}, safe(activate) = {}, safe(first^#) = {1}, 220.12/148.07 safe(c_4) = {}, safe(activate^#) = {}, safe(c_8) = {}, 220.12/148.07 safe(sel^#) = {2}, safe(c_9) = {} 220.12/148.07 220.12/148.07 and precedence 220.12/148.07 220.12/148.07 from ~ first, from ~ activate, from ~ sel^#, first ~ activate, 220.12/148.07 first ~ sel^#, activate ~ sel^#, first^# ~ activate^# . 220.12/148.07 220.12/148.07 Following symbols are considered recursive: 220.12/148.07 220.12/148.07 {from, first, activate, first^#, activate^#, sel^#} 220.12/148.07 220.12/148.07 The recursion depth is 1. 220.12/148.07 220.12/148.07 Further, following argument filtering is employed: 220.12/148.07 220.12/148.07 pi(from) = [], pi(cons) = [2], pi(n__from) = [], pi(s) = [1], 220.12/148.07 pi(first) = [], pi(0) = [], pi(nil) = [], pi(n__first) = [2], 220.12/148.07 pi(activate) = [], pi(first^#) = [2], pi(c_4) = [1], 220.12/148.07 pi(activate^#) = [1], pi(c_8) = [1], pi(sel^#) = [1], pi(c_9) = [1] 220.12/148.07 220.12/148.07 Usable defined function symbols are a subset of: 220.12/148.07 220.12/148.07 {first^#, activate^#, sel^#} 220.12/148.07 220.12/148.07 For your convenience, here are the satisfied ordering constraints: 220.12/148.07 220.12/148.07 pi(first^#(s(X), cons(Y, Z))) = first^#(cons(; Z);) 220.12/148.07 > c_4(activate^#(Z;);) 220.12/148.07 = pi(c_4(activate^#(Z))) 220.12/148.07 220.12/148.07 pi(activate^#(n__first(X1, X2))) = activate^#(n__first(; X2);) 220.12/148.07 > c_8(first^#(X2;);) 220.12/148.07 = pi(c_8(first^#(X1, X2))) 220.12/148.07 220.12/148.07 pi(sel^#(s(X), cons(Y, Z))) = sel^#(s(; X);) 220.12/148.07 > c_9(sel^#(X;);) 220.12/148.07 = pi(c_9(sel^#(X, activate(Z)))) 220.12/148.07 220.12/148.07 220.12/148.07 The strictly oriented rules are moved into the weak component. 220.12/148.07 220.12/148.07 We are left with following problem, upon which TcT provides the 220.12/148.07 certificate YES(O(1),O(1)). 220.12/148.07 220.12/148.07 Weak DPs: 220.12/148.07 { first^#(s(X), cons(Y, Z)) -> c_4(activate^#(Z)) 220.12/148.07 , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2)) 220.12/148.07 , sel^#(s(X), cons(Y, Z)) -> c_9(sel^#(X, activate(Z))) } 220.12/148.07 Weak Trs: 220.12/148.07 { from(X) -> cons(X, n__from(s(X))) 220.12/148.07 , from(X) -> n__from(X) 220.12/148.07 , first(X1, X2) -> n__first(X1, X2) 220.12/148.07 , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) 220.12/148.07 , first(0(), Z) -> nil() 220.12/148.07 , activate(X) -> X 220.12/148.07 , activate(n__from(X)) -> from(X) 220.12/148.07 , activate(n__first(X1, X2)) -> first(X1, X2) } 220.12/148.07 Obligation: 220.12/148.07 innermost runtime complexity 220.12/148.07 Answer: 220.12/148.07 YES(O(1),O(1)) 220.12/148.07 220.12/148.07 The following weak DPs constitute a sub-graph of the DG that is 220.12/148.07 closed under successors. The DPs are removed. 220.12/148.07 220.12/148.07 { first^#(s(X), cons(Y, Z)) -> c_4(activate^#(Z)) 220.12/148.07 , activate^#(n__first(X1, X2)) -> c_8(first^#(X1, X2)) 220.12/148.07 , sel^#(s(X), cons(Y, Z)) -> c_9(sel^#(X, activate(Z))) } 220.12/148.07 220.12/148.07 We are left with following problem, upon which TcT provides the 220.12/148.07 certificate YES(O(1),O(1)). 220.12/148.07 220.12/148.07 Weak Trs: 220.12/148.07 { from(X) -> cons(X, n__from(s(X))) 220.12/148.07 , from(X) -> n__from(X) 220.12/148.07 , first(X1, X2) -> n__first(X1, X2) 220.12/148.07 , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) 220.12/148.07 , first(0(), Z) -> nil() 220.12/148.07 , activate(X) -> X 220.12/148.07 , activate(n__from(X)) -> from(X) 220.12/148.07 , activate(n__first(X1, X2)) -> first(X1, X2) } 220.12/148.07 Obligation: 220.12/148.07 innermost runtime complexity 220.12/148.07 Answer: 220.12/148.07 YES(O(1),O(1)) 220.12/148.07 220.12/148.07 No rule is usable, rules are removed from the input problem. 220.12/148.07 220.12/148.07 We are left with following problem, upon which TcT provides the 220.12/148.07 certificate YES(O(1),O(1)). 220.12/148.07 220.12/148.07 Rules: Empty 220.12/148.07 Obligation: 220.12/148.07 innermost runtime complexity 220.12/148.07 Answer: 220.12/148.07 YES(O(1),O(1)) 220.12/148.07 220.12/148.07 Empty rules are trivially bounded 220.12/148.07 220.12/148.07 Hurray, we answered YES(O(1),O(n^2)) 220.12/148.09 EOF