YES(O(1),O(n^1)) 225.84/148.07 YES(O(1),O(n^1)) 225.84/148.07 225.84/148.07 We are left with following problem, upon which TcT provides the 225.84/148.07 certificate YES(O(1),O(n^1)). 225.84/148.07 225.84/148.07 Strict Trs: 225.84/148.07 { from(X) -> cons(X, n__from(s(X))) 225.84/148.07 , from(X) -> n__from(X) 225.84/148.07 , head(cons(X, XS)) -> X 225.84/148.07 , 2nd(cons(X, XS)) -> head(activate(XS)) 225.84/148.07 , activate(X) -> X 225.84/148.07 , activate(n__from(X)) -> from(X) 225.84/148.07 , activate(n__take(X1, X2)) -> take(X1, X2) 225.84/148.07 , take(X1, X2) -> n__take(X1, X2) 225.84/148.07 , take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) 225.84/148.07 , take(0(), XS) -> nil() 225.84/148.07 , sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) 225.84/148.07 , sel(0(), cons(X, XS)) -> X } 225.84/148.07 Obligation: 225.84/148.07 innermost runtime complexity 225.84/148.07 Answer: 225.84/148.07 YES(O(1),O(n^1)) 225.84/148.07 225.84/148.07 We add the following weak dependency pairs: 225.84/148.07 225.84/148.07 Strict DPs: 225.84/148.07 { from^#(X) -> c_1() 225.84/148.07 , from^#(X) -> c_2() 225.84/148.07 , head^#(cons(X, XS)) -> c_3() 225.84/148.07 , 2nd^#(cons(X, XS)) -> c_4(head^#(activate(XS))) 225.84/148.07 , activate^#(X) -> c_5() 225.84/148.07 , activate^#(n__from(X)) -> c_6(from^#(X)) 225.84/148.07 , activate^#(n__take(X1, X2)) -> c_7(take^#(X1, X2)) 225.84/148.07 , take^#(X1, X2) -> c_8() 225.84/148.07 , take^#(s(N), cons(X, XS)) -> c_9(activate^#(XS)) 225.84/148.07 , take^#(0(), XS) -> c_10() 225.84/148.07 , sel^#(s(N), cons(X, XS)) -> c_11(sel^#(N, activate(XS))) 225.84/148.07 , sel^#(0(), cons(X, XS)) -> c_12() } 225.84/148.07 225.84/148.07 and mark the set of starting terms. 225.84/148.07 225.84/148.07 We are left with following problem, upon which TcT provides the 225.84/148.07 certificate YES(O(1),O(n^1)). 225.84/148.07 225.84/148.07 Strict DPs: 225.84/148.07 { from^#(X) -> c_1() 225.84/148.07 , from^#(X) -> c_2() 225.84/148.07 , head^#(cons(X, XS)) -> c_3() 225.84/148.07 , 2nd^#(cons(X, XS)) -> c_4(head^#(activate(XS))) 225.84/148.07 , activate^#(X) -> c_5() 225.84/148.07 , activate^#(n__from(X)) -> c_6(from^#(X)) 225.84/148.07 , activate^#(n__take(X1, X2)) -> c_7(take^#(X1, X2)) 225.84/148.07 , take^#(X1, X2) -> c_8() 225.84/148.07 , take^#(s(N), cons(X, XS)) -> c_9(activate^#(XS)) 225.84/148.07 , take^#(0(), XS) -> c_10() 225.84/148.07 , sel^#(s(N), cons(X, XS)) -> c_11(sel^#(N, activate(XS))) 225.84/148.07 , sel^#(0(), cons(X, XS)) -> c_12() } 225.84/148.07 Strict Trs: 225.84/148.07 { from(X) -> cons(X, n__from(s(X))) 225.84/148.07 , from(X) -> n__from(X) 225.84/148.07 , head(cons(X, XS)) -> X 225.84/148.07 , 2nd(cons(X, XS)) -> head(activate(XS)) 225.84/148.07 , activate(X) -> X 225.84/148.07 , activate(n__from(X)) -> from(X) 225.84/148.07 , activate(n__take(X1, X2)) -> take(X1, X2) 225.84/148.07 , take(X1, X2) -> n__take(X1, X2) 225.84/148.07 , take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) 225.84/148.07 , take(0(), XS) -> nil() 225.84/148.07 , sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) 225.84/148.07 , sel(0(), cons(X, XS)) -> X } 225.84/148.07 Obligation: 225.84/148.07 innermost runtime complexity 225.84/148.07 Answer: 225.84/148.07 YES(O(1),O(n^1)) 225.84/148.07 225.84/148.07 We replace rewrite rules by usable rules: 225.84/148.07 225.84/148.07 Strict Usable Rules: 225.84/148.07 { from(X) -> cons(X, n__from(s(X))) 225.84/148.07 , from(X) -> n__from(X) 225.84/148.07 , activate(X) -> X 225.84/148.07 , activate(n__from(X)) -> from(X) 225.84/148.07 , activate(n__take(X1, X2)) -> take(X1, X2) 225.84/148.07 , take(X1, X2) -> n__take(X1, X2) 225.84/148.07 , take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) 225.84/148.07 , take(0(), XS) -> nil() } 225.84/148.07 225.84/148.07 We are left with following problem, upon which TcT provides the 225.84/148.07 certificate YES(O(1),O(n^1)). 225.84/148.07 225.84/148.07 Strict DPs: 225.84/148.07 { from^#(X) -> c_1() 225.84/148.07 , from^#(X) -> c_2() 225.84/148.07 , head^#(cons(X, XS)) -> c_3() 225.84/148.07 , 2nd^#(cons(X, XS)) -> c_4(head^#(activate(XS))) 225.84/148.07 , activate^#(X) -> c_5() 225.84/148.07 , activate^#(n__from(X)) -> c_6(from^#(X)) 225.84/148.07 , activate^#(n__take(X1, X2)) -> c_7(take^#(X1, X2)) 225.84/148.07 , take^#(X1, X2) -> c_8() 225.84/148.07 , take^#(s(N), cons(X, XS)) -> c_9(activate^#(XS)) 225.84/148.07 , take^#(0(), XS) -> c_10() 225.84/148.07 , sel^#(s(N), cons(X, XS)) -> c_11(sel^#(N, activate(XS))) 225.84/148.07 , sel^#(0(), cons(X, XS)) -> c_12() } 225.84/148.07 Strict Trs: 225.84/148.07 { from(X) -> cons(X, n__from(s(X))) 225.84/148.07 , from(X) -> n__from(X) 225.84/148.07 , activate(X) -> X 225.84/148.07 , activate(n__from(X)) -> from(X) 225.84/148.07 , activate(n__take(X1, X2)) -> take(X1, X2) 225.84/148.07 , take(X1, X2) -> n__take(X1, X2) 225.84/148.07 , take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) 225.84/148.07 , take(0(), XS) -> nil() } 225.84/148.07 Obligation: 225.84/148.07 innermost runtime complexity 225.84/148.07 Answer: 225.84/148.07 YES(O(1),O(n^1)) 225.84/148.07 225.84/148.07 The weightgap principle applies (using the following constant 225.84/148.07 growth matrix-interpretation) 225.84/148.07 225.84/148.07 The following argument positions are usable: 225.84/148.07 Uargs(cons) = {2}, Uargs(n__take) = {2}, Uargs(head^#) = {1}, 225.84/148.07 Uargs(c_4) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1}, 225.84/148.07 Uargs(c_9) = {1}, Uargs(sel^#) = {2}, Uargs(c_11) = {1} 225.84/148.07 225.84/148.07 TcT has computed the following constructor-restricted matrix 225.84/148.07 interpretation. 225.84/148.07 225.84/148.07 [from](x1) = [1] 225.84/148.07 [0] 225.84/148.07 225.84/148.07 [cons](x1, x2) = [1 0] x2 + [0] 225.84/148.07 [0 0] [0] 225.84/148.07 225.84/148.07 [n__from](x1) = [0] 225.84/148.07 [0] 225.84/148.07 225.84/148.07 [s](x1) = [1 0] x1 + [2] 225.84/148.07 [0 0] [0] 225.84/148.07 225.84/148.07 [activate](x1) = [1 0] x1 + [2] 225.84/148.07 [0 1] [0] 225.84/148.07 225.84/148.07 [take](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 225.84/148.07 [0 0] [0 0] [0] 225.84/148.07 225.84/148.07 [0] = [0] 225.84/148.07 [0] 225.84/148.07 225.84/148.07 [nil] = [0] 225.84/148.07 [0] 225.84/148.07 225.84/148.07 [n__take](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 225.84/148.07 [0 0] [0 0] [0] 225.84/148.07 225.84/148.07 [from^#](x1) = [0 0] x1 + [1] 225.84/148.07 [1 1] [1] 225.84/148.07 225.84/148.07 [c_1] = [0] 225.84/148.07 [1] 225.84/148.07 225.84/148.07 [c_2] = [1] 225.84/148.07 [1] 225.84/148.07 225.84/148.07 [head^#](x1) = [2 0] x1 + [2] 225.84/148.07 [0 0] [2] 225.84/148.07 225.84/148.07 [c_3] = [1] 225.84/148.07 [1] 225.84/148.07 225.84/148.07 [2nd^#](x1) = [2 2] x1 + [2] 225.84/148.07 [1 2] [2] 225.84/148.07 225.84/148.07 [c_4](x1) = [1 0] x1 + [1] 225.84/148.07 [0 1] [2] 225.84/148.07 225.84/148.07 [activate^#](x1) = [0 0] x1 + [2] 225.84/148.07 [1 1] [2] 225.84/148.07 225.84/148.07 [c_5] = [1] 225.84/148.07 [1] 225.84/148.07 225.84/148.07 [c_6](x1) = [1 0] x1 + [2] 225.84/148.07 [0 1] [2] 225.84/148.07 225.84/148.07 [c_7](x1) = [1 0] x1 + [1] 225.84/148.07 [0 1] [1] 225.84/148.07 225.84/148.07 [take^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2] 225.84/148.07 [1 1] [2 2] [2] 225.84/148.07 225.84/148.07 [c_8] = [1] 225.84/148.07 [1] 225.84/148.07 225.84/148.07 [c_9](x1) = [1 0] x1 + [1] 225.84/148.07 [0 1] [1] 225.84/148.07 225.84/148.07 [c_10] = [1] 225.84/148.07 [1] 225.84/148.07 225.84/148.07 [sel^#](x1, x2) = [2 0] x2 + [0] 225.84/148.07 [0 0] [0] 225.84/148.07 225.84/148.07 [c_11](x1) = [1 0] x1 + [2] 225.84/148.07 [0 1] [2] 225.84/148.07 225.84/148.07 [c_12] = [1] 225.84/148.07 [1] 225.84/148.07 225.84/148.07 The order satisfies the following ordering constraints: 225.84/148.07 225.84/148.07 [from(X)] = [1] 225.84/148.07 [0] 225.84/148.07 > [0] 225.84/148.07 [0] 225.84/148.07 = [cons(X, n__from(s(X)))] 225.84/148.07 225.84/148.07 [from(X)] = [1] 225.84/148.07 [0] 225.84/148.07 > [0] 225.84/148.07 [0] 225.84/148.07 = [n__from(X)] 225.84/148.07 225.84/148.07 [activate(X)] = [1 0] X + [2] 225.84/148.07 [0 1] [0] 225.84/148.07 > [1 0] X + [0] 225.84/148.07 [0 1] [0] 225.84/148.07 = [X] 225.84/148.07 225.84/148.07 [activate(n__from(X))] = [2] 225.84/148.07 [0] 225.84/148.07 > [1] 225.84/148.07 [0] 225.84/148.07 = [from(X)] 225.84/148.07 225.84/148.07 [activate(n__take(X1, X2))] = [1 0] X1 + [1 0] X2 + [2] 225.84/148.07 [0 0] [0 0] [0] 225.84/148.07 > [1 0] X1 + [1 0] X2 + [1] 225.84/148.07 [0 0] [0 0] [0] 225.84/148.07 = [take(X1, X2)] 225.84/148.07 225.84/148.07 [take(X1, X2)] = [1 0] X1 + [1 0] X2 + [1] 225.84/148.07 [0 0] [0 0] [0] 225.84/148.07 > [1 0] X1 + [1 0] X2 + [0] 225.84/148.07 [0 0] [0 0] [0] 225.84/148.07 = [n__take(X1, X2)] 225.84/148.07 225.84/148.07 [take(s(N), cons(X, XS))] = [1 0] XS + [1 0] N + [3] 225.84/148.07 [0 0] [0 0] [0] 225.84/148.07 > [1 0] XS + [1 0] N + [2] 225.84/148.07 [0 0] [0 0] [0] 225.84/148.07 = [cons(X, n__take(N, activate(XS)))] 225.84/148.07 225.84/148.07 [take(0(), XS)] = [1 0] XS + [1] 225.84/148.07 [0 0] [0] 225.84/148.07 > [0] 225.84/148.07 [0] 225.84/148.07 = [nil()] 225.84/148.07 225.84/148.07 [from^#(X)] = [0 0] X + [1] 225.84/148.07 [1 1] [1] 225.84/148.07 > [0] 225.84/148.07 [1] 225.84/148.07 = [c_1()] 225.84/148.07 225.84/148.07 [from^#(X)] = [0 0] X + [1] 225.84/148.07 [1 1] [1] 225.84/148.07 >= [1] 225.84/148.07 [1] 225.84/148.07 = [c_2()] 225.84/148.07 225.84/148.07 [head^#(cons(X, XS))] = [2 0] XS + [2] 225.84/148.07 [0 0] [2] 225.84/148.07 > [1] 225.84/148.07 [1] 225.84/148.07 = [c_3()] 225.84/148.07 225.84/148.07 [2nd^#(cons(X, XS))] = [2 0] XS + [2] 225.84/148.07 [1 0] [2] 225.84/148.07 ? [2 0] XS + [7] 225.84/148.07 [0 0] [4] 225.84/148.07 = [c_4(head^#(activate(XS)))] 225.84/148.07 225.84/148.07 [activate^#(X)] = [0 0] X + [2] 225.84/148.07 [1 1] [2] 225.84/148.07 > [1] 225.84/148.07 [1] 225.84/148.07 = [c_5()] 225.84/148.07 225.84/148.07 [activate^#(n__from(X))] = [2] 225.84/148.07 [2] 225.84/148.07 ? [0 0] X + [3] 225.84/148.07 [1 1] [3] 225.84/148.07 = [c_6(from^#(X))] 225.84/148.07 225.84/148.07 [activate^#(n__take(X1, X2))] = [0 0] X1 + [0 0] X2 + [2] 225.84/148.07 [1 0] [1 0] [2] 225.84/148.07 ? [0 0] X1 + [0 0] X2 + [3] 225.84/148.07 [1 1] [2 2] [3] 225.84/148.07 = [c_7(take^#(X1, X2))] 225.84/148.07 225.84/148.07 [take^#(X1, X2)] = [0 0] X1 + [0 0] X2 + [2] 225.84/148.07 [1 1] [2 2] [2] 225.84/148.07 > [1] 225.84/148.07 [1] 225.84/148.07 = [c_8()] 225.84/148.07 225.84/148.07 [take^#(s(N), cons(X, XS))] = [0 0] XS + [0 0] N + [2] 225.84/148.07 [2 0] [1 0] [4] 225.84/148.07 ? [0 0] XS + [3] 225.84/148.07 [1 1] [3] 225.84/148.07 = [c_9(activate^#(XS))] 225.84/148.07 225.84/148.07 [take^#(0(), XS)] = [0 0] XS + [2] 225.84/148.07 [2 2] [2] 225.84/148.07 > [1] 225.84/148.07 [1] 225.84/148.07 = [c_10()] 225.84/148.07 225.84/148.07 [sel^#(s(N), cons(X, XS))] = [2 0] XS + [0] 225.84/148.07 [0 0] [0] 225.84/148.07 ? [2 0] XS + [6] 225.84/148.07 [0 0] [2] 225.84/148.07 = [c_11(sel^#(N, activate(XS)))] 225.84/148.07 225.84/148.07 [sel^#(0(), cons(X, XS))] = [2 0] XS + [0] 225.84/148.07 [0 0] [0] 225.84/148.07 ? [1] 225.84/148.08 [1] 225.84/148.08 = [c_12()] 225.84/148.08 225.84/148.08 225.84/148.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 225.84/148.08 225.84/148.08 We are left with following problem, upon which TcT provides the 225.84/148.08 certificate YES(O(1),O(n^1)). 225.84/148.08 225.84/148.08 Strict DPs: 225.84/148.08 { from^#(X) -> c_2() 225.84/148.08 , 2nd^#(cons(X, XS)) -> c_4(head^#(activate(XS))) 225.84/148.08 , activate^#(n__from(X)) -> c_6(from^#(X)) 225.84/148.08 , activate^#(n__take(X1, X2)) -> c_7(take^#(X1, X2)) 225.84/148.08 , take^#(s(N), cons(X, XS)) -> c_9(activate^#(XS)) 225.84/148.08 , sel^#(s(N), cons(X, XS)) -> c_11(sel^#(N, activate(XS))) 225.84/148.08 , sel^#(0(), cons(X, XS)) -> c_12() } 225.84/148.08 Weak DPs: 225.84/148.08 { from^#(X) -> c_1() 225.84/148.08 , head^#(cons(X, XS)) -> c_3() 225.84/148.08 , activate^#(X) -> c_5() 225.84/148.08 , take^#(X1, X2) -> c_8() 225.84/148.08 , take^#(0(), XS) -> c_10() } 225.84/148.08 Weak Trs: 225.84/148.08 { from(X) -> cons(X, n__from(s(X))) 225.84/148.08 , from(X) -> n__from(X) 225.84/148.08 , activate(X) -> X 225.84/148.08 , activate(n__from(X)) -> from(X) 225.84/148.08 , activate(n__take(X1, X2)) -> take(X1, X2) 225.84/148.08 , take(X1, X2) -> n__take(X1, X2) 225.84/148.08 , take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) 225.84/148.08 , take(0(), XS) -> nil() } 225.84/148.08 Obligation: 225.84/148.08 innermost runtime complexity 225.84/148.08 Answer: 225.84/148.08 YES(O(1),O(n^1)) 225.84/148.08 225.84/148.08 We estimate the number of application of {1,2,7} by applications of 225.84/148.08 Pre({1,2,7}) = {3,6}. Here rules are labeled as follows: 225.84/148.08 225.84/148.08 DPs: 225.84/148.08 { 1: from^#(X) -> c_2() 225.84/148.08 , 2: 2nd^#(cons(X, XS)) -> c_4(head^#(activate(XS))) 225.84/148.08 , 3: activate^#(n__from(X)) -> c_6(from^#(X)) 225.84/148.08 , 4: activate^#(n__take(X1, X2)) -> c_7(take^#(X1, X2)) 225.84/148.08 , 5: take^#(s(N), cons(X, XS)) -> c_9(activate^#(XS)) 225.84/148.08 , 6: sel^#(s(N), cons(X, XS)) -> c_11(sel^#(N, activate(XS))) 225.84/148.08 , 7: sel^#(0(), cons(X, XS)) -> c_12() 225.84/148.08 , 8: from^#(X) -> c_1() 225.84/148.08 , 9: head^#(cons(X, XS)) -> c_3() 225.84/148.08 , 10: activate^#(X) -> c_5() 225.84/148.08 , 11: take^#(X1, X2) -> c_8() 225.84/148.08 , 12: take^#(0(), XS) -> c_10() } 225.84/148.08 225.84/148.08 We are left with following problem, upon which TcT provides the 225.84/148.08 certificate YES(O(1),O(n^1)). 225.84/148.08 225.84/148.08 Strict DPs: 225.84/148.08 { activate^#(n__from(X)) -> c_6(from^#(X)) 225.84/148.08 , activate^#(n__take(X1, X2)) -> c_7(take^#(X1, X2)) 225.84/148.08 , take^#(s(N), cons(X, XS)) -> c_9(activate^#(XS)) 225.84/148.08 , sel^#(s(N), cons(X, XS)) -> c_11(sel^#(N, activate(XS))) } 225.84/148.08 Weak DPs: 225.84/148.08 { from^#(X) -> c_1() 225.84/148.08 , from^#(X) -> c_2() 225.84/148.08 , head^#(cons(X, XS)) -> c_3() 225.84/148.08 , 2nd^#(cons(X, XS)) -> c_4(head^#(activate(XS))) 225.84/148.08 , activate^#(X) -> c_5() 225.84/148.08 , take^#(X1, X2) -> c_8() 225.84/148.08 , take^#(0(), XS) -> c_10() 225.84/148.08 , sel^#(0(), cons(X, XS)) -> c_12() } 225.84/148.08 Weak Trs: 225.84/148.08 { from(X) -> cons(X, n__from(s(X))) 225.84/148.08 , from(X) -> n__from(X) 225.84/148.08 , activate(X) -> X 225.84/148.08 , activate(n__from(X)) -> from(X) 225.84/148.08 , activate(n__take(X1, X2)) -> take(X1, X2) 225.84/148.08 , take(X1, X2) -> n__take(X1, X2) 225.84/148.08 , take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) 225.84/148.08 , take(0(), XS) -> nil() } 225.84/148.08 Obligation: 225.84/148.08 innermost runtime complexity 225.84/148.08 Answer: 225.84/148.08 YES(O(1),O(n^1)) 225.84/148.08 225.84/148.08 We estimate the number of application of {1} by applications of 225.84/148.08 Pre({1}) = {3}. Here rules are labeled as follows: 225.84/148.08 225.84/148.08 DPs: 225.84/148.08 { 1: activate^#(n__from(X)) -> c_6(from^#(X)) 225.84/148.08 , 2: activate^#(n__take(X1, X2)) -> c_7(take^#(X1, X2)) 225.84/148.08 , 3: take^#(s(N), cons(X, XS)) -> c_9(activate^#(XS)) 225.84/148.08 , 4: sel^#(s(N), cons(X, XS)) -> c_11(sel^#(N, activate(XS))) 225.84/148.08 , 5: from^#(X) -> c_1() 225.84/148.08 , 6: from^#(X) -> c_2() 225.84/148.08 , 7: head^#(cons(X, XS)) -> c_3() 225.84/148.08 , 8: 2nd^#(cons(X, XS)) -> c_4(head^#(activate(XS))) 225.84/148.08 , 9: activate^#(X) -> c_5() 225.84/148.08 , 10: take^#(X1, X2) -> c_8() 225.84/148.08 , 11: take^#(0(), XS) -> c_10() 225.84/148.08 , 12: sel^#(0(), cons(X, XS)) -> c_12() } 225.84/148.08 225.84/148.08 We are left with following problem, upon which TcT provides the 225.84/148.08 certificate YES(O(1),O(n^1)). 225.84/148.08 225.84/148.08 Strict DPs: 225.84/148.08 { activate^#(n__take(X1, X2)) -> c_7(take^#(X1, X2)) 225.84/148.08 , take^#(s(N), cons(X, XS)) -> c_9(activate^#(XS)) 225.84/148.08 , sel^#(s(N), cons(X, XS)) -> c_11(sel^#(N, activate(XS))) } 225.84/148.08 Weak DPs: 225.84/148.08 { from^#(X) -> c_1() 225.84/148.08 , from^#(X) -> c_2() 225.84/148.08 , head^#(cons(X, XS)) -> c_3() 225.84/148.08 , 2nd^#(cons(X, XS)) -> c_4(head^#(activate(XS))) 225.84/148.08 , activate^#(X) -> c_5() 225.84/148.08 , activate^#(n__from(X)) -> c_6(from^#(X)) 225.84/148.08 , take^#(X1, X2) -> c_8() 225.84/148.08 , take^#(0(), XS) -> c_10() 225.84/148.08 , sel^#(0(), cons(X, XS)) -> c_12() } 225.84/148.08 Weak Trs: 225.84/148.08 { from(X) -> cons(X, n__from(s(X))) 225.84/148.08 , from(X) -> n__from(X) 225.84/148.08 , activate(X) -> X 225.84/148.08 , activate(n__from(X)) -> from(X) 225.84/148.08 , activate(n__take(X1, X2)) -> take(X1, X2) 225.84/148.08 , take(X1, X2) -> n__take(X1, X2) 225.84/148.08 , take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) 225.84/148.08 , take(0(), XS) -> nil() } 225.84/148.08 Obligation: 225.84/148.08 innermost runtime complexity 225.84/148.08 Answer: 225.84/148.08 YES(O(1),O(n^1)) 225.84/148.08 225.84/148.08 The following weak DPs constitute a sub-graph of the DG that is 225.84/148.08 closed under successors. The DPs are removed. 225.84/148.08 225.84/148.08 { from^#(X) -> c_1() 225.84/148.08 , from^#(X) -> c_2() 225.84/148.08 , head^#(cons(X, XS)) -> c_3() 225.84/148.08 , 2nd^#(cons(X, XS)) -> c_4(head^#(activate(XS))) 225.84/148.08 , activate^#(X) -> c_5() 225.84/148.08 , activate^#(n__from(X)) -> c_6(from^#(X)) 225.84/148.08 , take^#(X1, X2) -> c_8() 225.84/148.08 , take^#(0(), XS) -> c_10() 225.84/148.08 , sel^#(0(), cons(X, XS)) -> c_12() } 225.84/148.08 225.84/148.08 We are left with following problem, upon which TcT provides the 225.84/148.08 certificate YES(O(1),O(n^1)). 225.84/148.08 225.84/148.08 Strict DPs: 225.84/148.08 { activate^#(n__take(X1, X2)) -> c_7(take^#(X1, X2)) 225.84/148.08 , take^#(s(N), cons(X, XS)) -> c_9(activate^#(XS)) 225.84/148.08 , sel^#(s(N), cons(X, XS)) -> c_11(sel^#(N, activate(XS))) } 225.84/148.08 Weak Trs: 225.84/148.08 { from(X) -> cons(X, n__from(s(X))) 225.84/148.08 , from(X) -> n__from(X) 225.84/148.08 , activate(X) -> X 225.84/148.08 , activate(n__from(X)) -> from(X) 225.84/148.08 , activate(n__take(X1, X2)) -> take(X1, X2) 225.84/148.08 , take(X1, X2) -> n__take(X1, X2) 225.84/148.08 , take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) 225.84/148.08 , take(0(), XS) -> nil() } 225.84/148.08 Obligation: 225.84/148.08 innermost runtime complexity 225.84/148.08 Answer: 225.84/148.08 YES(O(1),O(n^1)) 225.84/148.08 225.84/148.08 We use the processor 'Small Polynomial Path Order (PS,1-bounded)' 225.84/148.08 to orient following rules strictly. 225.84/148.08 225.84/148.08 DPs: 225.84/148.08 { 1: activate^#(n__take(X1, X2)) -> c_7(take^#(X1, X2)) 225.84/148.08 , 2: take^#(s(N), cons(X, XS)) -> c_9(activate^#(XS)) 225.84/148.08 , 3: sel^#(s(N), cons(X, XS)) -> c_11(sel^#(N, activate(XS))) } 225.84/148.08 Trs: 225.84/148.08 { from(X) -> n__from(X) 225.84/148.08 , take(0(), XS) -> nil() } 225.84/148.08 225.84/148.08 Sub-proof: 225.84/148.08 ---------- 225.84/148.08 The input was oriented with the instance of 'Small Polynomial Path 225.84/148.08 Order (PS,1-bounded)' as induced by the safe mapping 225.84/148.08 225.84/148.08 safe(from) = {1}, safe(cons) = {1, 2}, safe(n__from) = {1}, 225.84/148.08 safe(s) = {1}, safe(activate) = {}, safe(take) = {}, safe(0) = {}, 225.84/148.08 safe(nil) = {}, safe(n__take) = {1, 2}, safe(activate^#) = {}, 225.84/148.08 safe(c_7) = {}, safe(take^#) = {1}, safe(c_9) = {}, 225.84/148.08 safe(sel^#) = {2}, safe(c_11) = {} 225.84/148.08 225.84/148.08 and precedence 225.84/148.08 225.84/148.08 sel^# > from, from ~ activate, take ~ activate^#, take ~ take^#, 225.84/148.08 activate^# ~ take^# . 225.84/148.08 225.84/148.08 Following symbols are considered recursive: 225.84/148.08 225.84/148.08 {activate^#, take^#, sel^#} 225.84/148.08 225.84/148.08 The recursion depth is 1. 225.84/148.08 225.84/148.08 Further, following argument filtering is employed: 225.84/148.08 225.84/148.08 pi(from) = [], pi(cons) = [2], pi(n__from) = [], pi(s) = [1], 225.84/148.08 pi(activate) = [], pi(take) = [], pi(0) = [], pi(nil) = [], 225.84/148.08 pi(n__take) = [2], pi(activate^#) = [1], pi(c_7) = [1], 225.84/148.08 pi(take^#) = [2], pi(c_9) = [1], pi(sel^#) = [1], pi(c_11) = [1] 225.84/148.08 225.84/148.08 Usable defined function symbols are a subset of: 225.84/148.08 225.84/148.08 {activate^#, take^#, sel^#} 225.84/148.08 225.84/148.08 For your convenience, here are the satisfied ordering constraints: 225.84/148.08 225.84/148.08 pi(activate^#(n__take(X1, X2))) = activate^#(n__take(; X2);) 225.84/148.08 > c_7(take^#(X2;);) 225.84/148.08 = pi(c_7(take^#(X1, X2))) 225.84/148.08 225.84/148.08 pi(take^#(s(N), cons(X, XS))) = take^#(cons(; XS);) 225.84/148.08 > c_9(activate^#(XS;);) 225.84/148.08 = pi(c_9(activate^#(XS))) 225.84/148.08 225.84/148.08 pi(sel^#(s(N), cons(X, XS))) = sel^#(s(; N);) 225.84/148.08 > c_11(sel^#(N;);) 225.84/148.08 = pi(c_11(sel^#(N, activate(XS)))) 225.84/148.08 225.84/148.08 225.84/148.08 The strictly oriented rules are moved into the weak component. 225.84/148.08 225.84/148.08 We are left with following problem, upon which TcT provides the 225.84/148.08 certificate YES(O(1),O(1)). 225.84/148.08 225.84/148.08 Weak DPs: 225.84/148.08 { activate^#(n__take(X1, X2)) -> c_7(take^#(X1, X2)) 225.84/148.08 , take^#(s(N), cons(X, XS)) -> c_9(activate^#(XS)) 225.84/148.08 , sel^#(s(N), cons(X, XS)) -> c_11(sel^#(N, activate(XS))) } 225.84/148.08 Weak Trs: 225.84/148.08 { from(X) -> cons(X, n__from(s(X))) 225.84/148.08 , from(X) -> n__from(X) 225.84/148.08 , activate(X) -> X 225.84/148.08 , activate(n__from(X)) -> from(X) 225.84/148.08 , activate(n__take(X1, X2)) -> take(X1, X2) 225.84/148.08 , take(X1, X2) -> n__take(X1, X2) 225.84/148.08 , take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) 225.84/148.08 , take(0(), XS) -> nil() } 225.84/148.08 Obligation: 225.84/148.08 innermost runtime complexity 225.84/148.08 Answer: 225.84/148.08 YES(O(1),O(1)) 225.84/148.08 225.84/148.08 The following weak DPs constitute a sub-graph of the DG that is 225.84/148.08 closed under successors. The DPs are removed. 225.84/148.08 225.84/148.08 { activate^#(n__take(X1, X2)) -> c_7(take^#(X1, X2)) 225.84/148.08 , take^#(s(N), cons(X, XS)) -> c_9(activate^#(XS)) 225.84/148.08 , sel^#(s(N), cons(X, XS)) -> c_11(sel^#(N, activate(XS))) } 225.84/148.08 225.84/148.08 We are left with following problem, upon which TcT provides the 225.84/148.08 certificate YES(O(1),O(1)). 225.84/148.08 225.84/148.08 Weak Trs: 225.84/148.08 { from(X) -> cons(X, n__from(s(X))) 225.84/148.08 , from(X) -> n__from(X) 225.84/148.08 , activate(X) -> X 225.84/148.08 , activate(n__from(X)) -> from(X) 225.84/148.08 , activate(n__take(X1, X2)) -> take(X1, X2) 225.84/148.08 , take(X1, X2) -> n__take(X1, X2) 225.84/148.08 , take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) 225.84/148.08 , take(0(), XS) -> nil() } 225.84/148.08 Obligation: 225.84/148.08 innermost runtime complexity 225.84/148.08 Answer: 225.84/148.08 YES(O(1),O(1)) 225.84/148.08 225.84/148.08 No rule is usable, rules are removed from the input problem. 225.84/148.08 225.84/148.08 We are left with following problem, upon which TcT provides the 225.84/148.08 certificate YES(O(1),O(1)). 225.84/148.08 225.84/148.08 Rules: Empty 225.84/148.08 Obligation: 225.84/148.08 innermost runtime complexity 225.84/148.08 Answer: 225.84/148.08 YES(O(1),O(1)) 225.84/148.08 225.84/148.08 Empty rules are trivially bounded 225.84/148.08 225.84/148.08 Hurray, we answered YES(O(1),O(n^1)) 225.84/148.09 EOF