YES(O(1),O(n^3)) 473.91/297.08 YES(O(1),O(n^3)) 473.91/297.08 473.91/297.08 We are left with following problem, upon which TcT provides the 473.91/297.08 certificate YES(O(1),O(n^3)). 473.91/297.08 473.91/297.08 Strict Trs: 473.91/297.08 { active(f(0())) -> mark(cons(0(), f(s(0())))) 473.91/297.08 , active(f(s(0()))) -> mark(f(p(s(0())))) 473.91/297.08 , active(p(s(0()))) -> mark(0()) 473.91/297.08 , f(active(X)) -> f(X) 473.91/297.08 , f(mark(X)) -> f(X) 473.91/297.08 , mark(f(X)) -> active(f(mark(X))) 473.91/297.08 , mark(0()) -> active(0()) 473.91/297.08 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) 473.91/297.08 , mark(s(X)) -> active(s(mark(X))) 473.91/297.08 , mark(p(X)) -> active(p(mark(X))) 473.91/297.08 , cons(X1, active(X2)) -> cons(X1, X2) 473.91/297.08 , cons(X1, mark(X2)) -> cons(X1, X2) 473.91/297.08 , cons(active(X1), X2) -> cons(X1, X2) 473.91/297.08 , cons(mark(X1), X2) -> cons(X1, X2) 473.91/297.08 , s(active(X)) -> s(X) 473.91/297.08 , s(mark(X)) -> s(X) 473.91/297.08 , p(active(X)) -> p(X) 473.91/297.08 , p(mark(X)) -> p(X) } 473.91/297.08 Obligation: 473.91/297.08 derivational complexity 473.91/297.08 Answer: 473.91/297.08 YES(O(1),O(n^3)) 473.91/297.08 473.91/297.08 The weightgap principle applies (using the following nonconstant 473.91/297.08 growth matrix-interpretation) 473.91/297.08 473.91/297.08 TcT has computed the following triangular matrix interpretation. 473.91/297.08 Note that the diagonal of the component-wise maxima of 473.91/297.08 interpretation-entries contains no more than 1 non-zero entries. 473.91/297.08 473.91/297.08 [active](x1) = [1] x1 + [0] 473.91/297.08 473.91/297.08 [f](x1) = [1] x1 + [0] 473.91/297.08 473.91/297.08 [0] = [0] 473.91/297.08 473.91/297.08 [mark](x1) = [1] x1 + [0] 473.91/297.08 473.91/297.08 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 473.91/297.08 473.91/297.08 [s](x1) = [1] x1 + [1] 473.91/297.08 473.91/297.08 [p](x1) = [1] x1 + [0] 473.91/297.08 473.91/297.08 The order satisfies the following ordering constraints: 473.91/297.08 473.91/297.08 [active(f(0()))] = [0] 473.91/297.08 ? [1] 473.91/297.08 = [mark(cons(0(), f(s(0()))))] 473.91/297.08 473.91/297.08 [active(f(s(0())))] = [1] 473.91/297.08 >= [1] 473.91/297.08 = [mark(f(p(s(0()))))] 473.91/297.08 473.91/297.08 [active(p(s(0())))] = [1] 473.91/297.08 > [0] 473.91/297.08 = [mark(0())] 473.91/297.08 473.91/297.08 [f(active(X))] = [1] X + [0] 473.91/297.08 >= [1] X + [0] 473.91/297.08 = [f(X)] 473.91/297.08 473.91/297.08 [f(mark(X))] = [1] X + [0] 473.91/297.08 >= [1] X + [0] 473.91/297.08 = [f(X)] 473.91/297.08 473.91/297.08 [mark(f(X))] = [1] X + [0] 473.91/297.08 >= [1] X + [0] 473.91/297.08 = [active(f(mark(X)))] 473.91/297.08 473.91/297.08 [mark(0())] = [0] 473.91/297.08 >= [0] 473.91/297.08 = [active(0())] 473.91/297.08 473.91/297.08 [mark(cons(X1, X2))] = [1] X1 + [1] X2 + [0] 473.91/297.08 >= [1] X1 + [1] X2 + [0] 473.91/297.08 = [active(cons(mark(X1), X2))] 473.91/297.08 473.91/297.08 [mark(s(X))] = [1] X + [1] 473.91/297.08 >= [1] X + [1] 473.91/297.08 = [active(s(mark(X)))] 473.91/297.08 473.91/297.08 [mark(p(X))] = [1] X + [0] 473.91/297.08 >= [1] X + [0] 473.91/297.08 = [active(p(mark(X)))] 473.91/297.08 473.91/297.08 [cons(X1, active(X2))] = [1] X1 + [1] X2 + [0] 473.91/297.08 >= [1] X1 + [1] X2 + [0] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [cons(X1, mark(X2))] = [1] X1 + [1] X2 + [0] 473.91/297.08 >= [1] X1 + [1] X2 + [0] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [cons(active(X1), X2)] = [1] X1 + [1] X2 + [0] 473.91/297.08 >= [1] X1 + [1] X2 + [0] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [cons(mark(X1), X2)] = [1] X1 + [1] X2 + [0] 473.91/297.08 >= [1] X1 + [1] X2 + [0] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [s(active(X))] = [1] X + [1] 473.91/297.08 >= [1] X + [1] 473.91/297.08 = [s(X)] 473.91/297.08 473.91/297.08 [s(mark(X))] = [1] X + [1] 473.91/297.08 >= [1] X + [1] 473.91/297.08 = [s(X)] 473.91/297.08 473.91/297.08 [p(active(X))] = [1] X + [0] 473.91/297.08 >= [1] X + [0] 473.91/297.08 = [p(X)] 473.91/297.08 473.91/297.08 [p(mark(X))] = [1] X + [0] 473.91/297.08 >= [1] X + [0] 473.91/297.08 = [p(X)] 473.91/297.08 473.91/297.08 473.91/297.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 473.91/297.08 473.91/297.08 We are left with following problem, upon which TcT provides the 473.91/297.08 certificate YES(O(1),O(n^3)). 473.91/297.08 473.91/297.08 Strict Trs: 473.91/297.08 { active(f(0())) -> mark(cons(0(), f(s(0())))) 473.91/297.08 , active(f(s(0()))) -> mark(f(p(s(0())))) 473.91/297.08 , f(active(X)) -> f(X) 473.91/297.08 , f(mark(X)) -> f(X) 473.91/297.08 , mark(f(X)) -> active(f(mark(X))) 473.91/297.08 , mark(0()) -> active(0()) 473.91/297.08 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) 473.91/297.08 , mark(s(X)) -> active(s(mark(X))) 473.91/297.08 , mark(p(X)) -> active(p(mark(X))) 473.91/297.08 , cons(X1, active(X2)) -> cons(X1, X2) 473.91/297.08 , cons(X1, mark(X2)) -> cons(X1, X2) 473.91/297.08 , cons(active(X1), X2) -> cons(X1, X2) 473.91/297.08 , cons(mark(X1), X2) -> cons(X1, X2) 473.91/297.08 , s(active(X)) -> s(X) 473.91/297.08 , s(mark(X)) -> s(X) 473.91/297.08 , p(active(X)) -> p(X) 473.91/297.08 , p(mark(X)) -> p(X) } 473.91/297.08 Weak Trs: { active(p(s(0()))) -> mark(0()) } 473.91/297.08 Obligation: 473.91/297.08 derivational complexity 473.91/297.08 Answer: 473.91/297.08 YES(O(1),O(n^3)) 473.91/297.08 473.91/297.08 The weightgap principle applies (using the following nonconstant 473.91/297.08 growth matrix-interpretation) 473.91/297.08 473.91/297.08 TcT has computed the following triangular matrix interpretation. 473.91/297.08 Note that the diagonal of the component-wise maxima of 473.91/297.08 interpretation-entries contains no more than 1 non-zero entries. 473.91/297.08 473.91/297.08 [active](x1) = [1] x1 + [1] 473.91/297.08 473.91/297.08 [f](x1) = [1] x1 + [0] 473.91/297.08 473.91/297.08 [0] = [0] 473.91/297.08 473.91/297.08 [mark](x1) = [1] x1 + [0] 473.91/297.08 473.91/297.08 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 473.91/297.08 473.91/297.08 [s](x1) = [1] x1 + [0] 473.91/297.08 473.91/297.08 [p](x1) = [1] x1 + [0] 473.91/297.08 473.91/297.08 The order satisfies the following ordering constraints: 473.91/297.08 473.91/297.08 [active(f(0()))] = [1] 473.91/297.08 > [0] 473.91/297.08 = [mark(cons(0(), f(s(0()))))] 473.91/297.08 473.91/297.08 [active(f(s(0())))] = [1] 473.91/297.08 > [0] 473.91/297.08 = [mark(f(p(s(0()))))] 473.91/297.08 473.91/297.08 [active(p(s(0())))] = [1] 473.91/297.08 > [0] 473.91/297.08 = [mark(0())] 473.91/297.08 473.91/297.08 [f(active(X))] = [1] X + [1] 473.91/297.08 > [1] X + [0] 473.91/297.08 = [f(X)] 473.91/297.08 473.91/297.08 [f(mark(X))] = [1] X + [0] 473.91/297.08 >= [1] X + [0] 473.91/297.08 = [f(X)] 473.91/297.08 473.91/297.08 [mark(f(X))] = [1] X + [0] 473.91/297.08 ? [1] X + [1] 473.91/297.08 = [active(f(mark(X)))] 473.91/297.08 473.91/297.08 [mark(0())] = [0] 473.91/297.08 ? [1] 473.91/297.08 = [active(0())] 473.91/297.08 473.91/297.08 [mark(cons(X1, X2))] = [1] X1 + [1] X2 + [0] 473.91/297.08 ? [1] X1 + [1] X2 + [1] 473.91/297.08 = [active(cons(mark(X1), X2))] 473.91/297.08 473.91/297.08 [mark(s(X))] = [1] X + [0] 473.91/297.08 ? [1] X + [1] 473.91/297.08 = [active(s(mark(X)))] 473.91/297.08 473.91/297.08 [mark(p(X))] = [1] X + [0] 473.91/297.08 ? [1] X + [1] 473.91/297.08 = [active(p(mark(X)))] 473.91/297.08 473.91/297.08 [cons(X1, active(X2))] = [1] X1 + [1] X2 + [1] 473.91/297.08 > [1] X1 + [1] X2 + [0] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [cons(X1, mark(X2))] = [1] X1 + [1] X2 + [0] 473.91/297.08 >= [1] X1 + [1] X2 + [0] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [cons(active(X1), X2)] = [1] X1 + [1] X2 + [1] 473.91/297.08 > [1] X1 + [1] X2 + [0] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [cons(mark(X1), X2)] = [1] X1 + [1] X2 + [0] 473.91/297.08 >= [1] X1 + [1] X2 + [0] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [s(active(X))] = [1] X + [1] 473.91/297.08 > [1] X + [0] 473.91/297.08 = [s(X)] 473.91/297.08 473.91/297.08 [s(mark(X))] = [1] X + [0] 473.91/297.08 >= [1] X + [0] 473.91/297.08 = [s(X)] 473.91/297.08 473.91/297.08 [p(active(X))] = [1] X + [1] 473.91/297.08 > [1] X + [0] 473.91/297.08 = [p(X)] 473.91/297.08 473.91/297.08 [p(mark(X))] = [1] X + [0] 473.91/297.08 >= [1] X + [0] 473.91/297.08 = [p(X)] 473.91/297.08 473.91/297.08 473.91/297.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 473.91/297.08 473.91/297.08 We are left with following problem, upon which TcT provides the 473.91/297.08 certificate YES(O(1),O(n^3)). 473.91/297.08 473.91/297.08 Strict Trs: 473.91/297.08 { f(mark(X)) -> f(X) 473.91/297.08 , mark(f(X)) -> active(f(mark(X))) 473.91/297.08 , mark(0()) -> active(0()) 473.91/297.08 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) 473.91/297.08 , mark(s(X)) -> active(s(mark(X))) 473.91/297.08 , mark(p(X)) -> active(p(mark(X))) 473.91/297.08 , cons(X1, mark(X2)) -> cons(X1, X2) 473.91/297.08 , cons(mark(X1), X2) -> cons(X1, X2) 473.91/297.08 , s(mark(X)) -> s(X) 473.91/297.08 , p(mark(X)) -> p(X) } 473.91/297.08 Weak Trs: 473.91/297.08 { active(f(0())) -> mark(cons(0(), f(s(0())))) 473.91/297.08 , active(f(s(0()))) -> mark(f(p(s(0())))) 473.91/297.08 , active(p(s(0()))) -> mark(0()) 473.91/297.08 , f(active(X)) -> f(X) 473.91/297.08 , cons(X1, active(X2)) -> cons(X1, X2) 473.91/297.08 , cons(active(X1), X2) -> cons(X1, X2) 473.91/297.08 , s(active(X)) -> s(X) 473.91/297.08 , p(active(X)) -> p(X) } 473.91/297.08 Obligation: 473.91/297.08 derivational complexity 473.91/297.08 Answer: 473.91/297.08 YES(O(1),O(n^3)) 473.91/297.08 473.91/297.08 The weightgap principle applies (using the following nonconstant 473.91/297.08 growth matrix-interpretation) 473.91/297.08 473.91/297.08 TcT has computed the following triangular matrix interpretation. 473.91/297.08 Note that the diagonal of the component-wise maxima of 473.91/297.08 interpretation-entries contains no more than 1 non-zero entries. 473.91/297.08 473.91/297.08 [active](x1) = [1] x1 + [1] 473.91/297.08 473.91/297.08 [f](x1) = [1] x1 + [0] 473.91/297.08 473.91/297.08 [0] = [0] 473.91/297.08 473.91/297.08 [mark](x1) = [1] x1 + [1] 473.91/297.08 473.91/297.08 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 473.91/297.08 473.91/297.08 [s](x1) = [1] x1 + [0] 473.91/297.08 473.91/297.08 [p](x1) = [1] x1 + [0] 473.91/297.08 473.91/297.08 The order satisfies the following ordering constraints: 473.91/297.08 473.91/297.08 [active(f(0()))] = [1] 473.91/297.08 >= [1] 473.91/297.08 = [mark(cons(0(), f(s(0()))))] 473.91/297.08 473.91/297.08 [active(f(s(0())))] = [1] 473.91/297.08 >= [1] 473.91/297.08 = [mark(f(p(s(0()))))] 473.91/297.08 473.91/297.08 [active(p(s(0())))] = [1] 473.91/297.08 >= [1] 473.91/297.08 = [mark(0())] 473.91/297.08 473.91/297.08 [f(active(X))] = [1] X + [1] 473.91/297.08 > [1] X + [0] 473.91/297.08 = [f(X)] 473.91/297.08 473.91/297.08 [f(mark(X))] = [1] X + [1] 473.91/297.08 > [1] X + [0] 473.91/297.08 = [f(X)] 473.91/297.08 473.91/297.08 [mark(f(X))] = [1] X + [1] 473.91/297.08 ? [1] X + [2] 473.91/297.08 = [active(f(mark(X)))] 473.91/297.08 473.91/297.08 [mark(0())] = [1] 473.91/297.08 >= [1] 473.91/297.08 = [active(0())] 473.91/297.08 473.91/297.08 [mark(cons(X1, X2))] = [1] X1 + [1] X2 + [1] 473.91/297.08 ? [1] X1 + [1] X2 + [2] 473.91/297.08 = [active(cons(mark(X1), X2))] 473.91/297.08 473.91/297.08 [mark(s(X))] = [1] X + [1] 473.91/297.08 ? [1] X + [2] 473.91/297.08 = [active(s(mark(X)))] 473.91/297.08 473.91/297.08 [mark(p(X))] = [1] X + [1] 473.91/297.08 ? [1] X + [2] 473.91/297.08 = [active(p(mark(X)))] 473.91/297.08 473.91/297.08 [cons(X1, active(X2))] = [1] X1 + [1] X2 + [1] 473.91/297.08 > [1] X1 + [1] X2 + [0] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [cons(X1, mark(X2))] = [1] X1 + [1] X2 + [1] 473.91/297.08 > [1] X1 + [1] X2 + [0] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [cons(active(X1), X2)] = [1] X1 + [1] X2 + [1] 473.91/297.08 > [1] X1 + [1] X2 + [0] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [cons(mark(X1), X2)] = [1] X1 + [1] X2 + [1] 473.91/297.08 > [1] X1 + [1] X2 + [0] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [s(active(X))] = [1] X + [1] 473.91/297.08 > [1] X + [0] 473.91/297.08 = [s(X)] 473.91/297.08 473.91/297.08 [s(mark(X))] = [1] X + [1] 473.91/297.08 > [1] X + [0] 473.91/297.08 = [s(X)] 473.91/297.08 473.91/297.08 [p(active(X))] = [1] X + [1] 473.91/297.08 > [1] X + [0] 473.91/297.08 = [p(X)] 473.91/297.08 473.91/297.08 [p(mark(X))] = [1] X + [1] 473.91/297.08 > [1] X + [0] 473.91/297.08 = [p(X)] 473.91/297.08 473.91/297.08 473.91/297.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 473.91/297.08 473.91/297.08 We are left with following problem, upon which TcT provides the 473.91/297.08 certificate YES(O(1),O(n^3)). 473.91/297.08 473.91/297.08 Strict Trs: 473.91/297.08 { mark(f(X)) -> active(f(mark(X))) 473.91/297.08 , mark(0()) -> active(0()) 473.91/297.08 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) 473.91/297.08 , mark(s(X)) -> active(s(mark(X))) 473.91/297.08 , mark(p(X)) -> active(p(mark(X))) } 473.91/297.08 Weak Trs: 473.91/297.08 { active(f(0())) -> mark(cons(0(), f(s(0())))) 473.91/297.08 , active(f(s(0()))) -> mark(f(p(s(0())))) 473.91/297.08 , active(p(s(0()))) -> mark(0()) 473.91/297.08 , f(active(X)) -> f(X) 473.91/297.08 , f(mark(X)) -> f(X) 473.91/297.08 , cons(X1, active(X2)) -> cons(X1, X2) 473.91/297.08 , cons(X1, mark(X2)) -> cons(X1, X2) 473.91/297.08 , cons(active(X1), X2) -> cons(X1, X2) 473.91/297.08 , cons(mark(X1), X2) -> cons(X1, X2) 473.91/297.08 , s(active(X)) -> s(X) 473.91/297.08 , s(mark(X)) -> s(X) 473.91/297.08 , p(active(X)) -> p(X) 473.91/297.08 , p(mark(X)) -> p(X) } 473.91/297.08 Obligation: 473.91/297.08 derivational complexity 473.91/297.08 Answer: 473.91/297.08 YES(O(1),O(n^3)) 473.91/297.08 473.91/297.08 The weightgap principle applies (using the following nonconstant 473.91/297.08 growth matrix-interpretation) 473.91/297.08 473.91/297.08 TcT has computed the following triangular matrix interpretation. 473.91/297.08 Note that the diagonal of the component-wise maxima of 473.91/297.08 interpretation-entries contains no more than 1 non-zero entries. 473.91/297.08 473.91/297.08 [active](x1) = [1 1] x1 + [0] 473.91/297.08 [0 0] [0] 473.91/297.08 473.91/297.08 [f](x1) = [1 0] x1 + [0] 473.91/297.08 [0 0] [2] 473.91/297.08 473.91/297.08 [0] = [0] 473.91/297.08 [0] 473.91/297.08 473.91/297.08 [mark](x1) = [1 0] x1 + [2] 473.91/297.08 [0 0] [0] 473.91/297.08 473.91/297.08 [cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 473.91/297.08 [0 0] [0 0] [2] 473.91/297.08 473.91/297.08 [s](x1) = [1 0] x1 + [0] 473.91/297.08 [0 0] [0] 473.91/297.08 473.91/297.08 [p](x1) = [1 0] x1 + [0] 473.91/297.08 [0 0] [2] 473.91/297.08 473.91/297.08 The order satisfies the following ordering constraints: 473.91/297.08 473.91/297.08 [active(f(0()))] = [2] 473.91/297.08 [0] 473.91/297.08 >= [2] 473.91/297.08 [0] 473.91/297.08 = [mark(cons(0(), f(s(0()))))] 473.91/297.08 473.91/297.08 [active(f(s(0())))] = [2] 473.91/297.08 [0] 473.91/297.08 >= [2] 473.91/297.08 [0] 473.91/297.08 = [mark(f(p(s(0()))))] 473.91/297.08 473.91/297.08 [active(p(s(0())))] = [2] 473.91/297.08 [0] 473.91/297.08 >= [2] 473.91/297.08 [0] 473.91/297.08 = [mark(0())] 473.91/297.08 473.91/297.08 [f(active(X))] = [1 1] X + [0] 473.91/297.08 [0 0] [2] 473.91/297.08 >= [1 0] X + [0] 473.91/297.08 [0 0] [2] 473.91/297.08 = [f(X)] 473.91/297.08 473.91/297.08 [f(mark(X))] = [1 0] X + [2] 473.91/297.08 [0 0] [2] 473.91/297.08 > [1 0] X + [0] 473.91/297.08 [0 0] [2] 473.91/297.08 = [f(X)] 473.91/297.08 473.91/297.08 [mark(f(X))] = [1 0] X + [2] 473.91/297.08 [0 0] [0] 473.91/297.08 ? [1 0] X + [4] 473.91/297.08 [0 0] [0] 473.91/297.08 = [active(f(mark(X)))] 473.91/297.08 473.91/297.08 [mark(0())] = [2] 473.91/297.08 [0] 473.91/297.08 > [0] 473.91/297.08 [0] 473.91/297.08 = [active(0())] 473.91/297.08 473.91/297.08 [mark(cons(X1, X2))] = [1 0] X1 + [1 0] X2 + [2] 473.91/297.08 [0 0] [0 0] [0] 473.91/297.08 ? [1 0] X1 + [1 0] X2 + [4] 473.91/297.08 [0 0] [0 0] [0] 473.91/297.08 = [active(cons(mark(X1), X2))] 473.91/297.08 473.91/297.08 [mark(s(X))] = [1 0] X + [2] 473.91/297.08 [0 0] [0] 473.91/297.08 >= [1 0] X + [2] 473.91/297.08 [0 0] [0] 473.91/297.08 = [active(s(mark(X)))] 473.91/297.08 473.91/297.08 [mark(p(X))] = [1 0] X + [2] 473.91/297.08 [0 0] [0] 473.91/297.08 ? [1 0] X + [4] 473.91/297.08 [0 0] [0] 473.91/297.08 = [active(p(mark(X)))] 473.91/297.08 473.91/297.08 [cons(X1, active(X2))] = [1 0] X1 + [1 1] X2 + [0] 473.91/297.08 [0 0] [0 0] [2] 473.91/297.08 >= [1 0] X1 + [1 0] X2 + [0] 473.91/297.08 [0 0] [0 0] [2] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [cons(X1, mark(X2))] = [1 0] X1 + [1 0] X2 + [2] 473.91/297.08 [0 0] [0 0] [2] 473.91/297.08 > [1 0] X1 + [1 0] X2 + [0] 473.91/297.08 [0 0] [0 0] [2] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [cons(active(X1), X2)] = [1 1] X1 + [1 0] X2 + [0] 473.91/297.08 [0 0] [0 0] [2] 473.91/297.08 >= [1 0] X1 + [1 0] X2 + [0] 473.91/297.08 [0 0] [0 0] [2] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [cons(mark(X1), X2)] = [1 0] X1 + [1 0] X2 + [2] 473.91/297.08 [0 0] [0 0] [2] 473.91/297.08 > [1 0] X1 + [1 0] X2 + [0] 473.91/297.08 [0 0] [0 0] [2] 473.91/297.08 = [cons(X1, X2)] 473.91/297.08 473.91/297.08 [s(active(X))] = [1 1] X + [0] 473.91/297.08 [0 0] [0] 473.91/297.08 >= [1 0] X + [0] 473.91/297.08 [0 0] [0] 473.91/297.08 = [s(X)] 473.91/297.08 473.91/297.08 [s(mark(X))] = [1 0] X + [2] 473.91/297.08 [0 0] [0] 473.91/297.08 > [1 0] X + [0] 473.91/297.08 [0 0] [0] 473.91/297.08 = [s(X)] 473.91/297.08 473.91/297.08 [p(active(X))] = [1 1] X + [0] 473.91/297.08 [0 0] [2] 473.91/297.08 >= [1 0] X + [0] 473.91/297.08 [0 0] [2] 473.91/297.08 = [p(X)] 473.91/297.08 473.91/297.08 [p(mark(X))] = [1 0] X + [2] 473.91/297.08 [0 0] [2] 473.91/297.08 > [1 0] X + [0] 473.91/297.08 [0 0] [2] 473.91/297.08 = [p(X)] 473.91/297.08 473.91/297.08 473.91/297.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 473.91/297.08 473.91/297.08 We are left with following problem, upon which TcT provides the 473.91/297.08 certificate YES(O(1),O(n^3)). 473.91/297.08 473.91/297.08 Strict Trs: 473.91/297.08 { mark(f(X)) -> active(f(mark(X))) 473.91/297.08 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) 473.91/297.08 , mark(s(X)) -> active(s(mark(X))) 473.91/297.08 , mark(p(X)) -> active(p(mark(X))) } 473.91/297.08 Weak Trs: 473.91/297.08 { active(f(0())) -> mark(cons(0(), f(s(0())))) 473.91/297.08 , active(f(s(0()))) -> mark(f(p(s(0())))) 473.91/297.08 , active(p(s(0()))) -> mark(0()) 473.91/297.08 , f(active(X)) -> f(X) 473.91/297.08 , f(mark(X)) -> f(X) 473.91/297.08 , mark(0()) -> active(0()) 473.91/297.08 , cons(X1, active(X2)) -> cons(X1, X2) 473.91/297.08 , cons(X1, mark(X2)) -> cons(X1, X2) 473.91/297.08 , cons(active(X1), X2) -> cons(X1, X2) 473.91/297.08 , cons(mark(X1), X2) -> cons(X1, X2) 473.91/297.08 , s(active(X)) -> s(X) 473.91/297.08 , s(mark(X)) -> s(X) 473.91/297.08 , p(active(X)) -> p(X) 473.91/297.08 , p(mark(X)) -> p(X) } 473.91/297.08 Obligation: 473.91/297.08 derivational complexity 473.91/297.08 Answer: 473.91/297.08 YES(O(1),O(n^3)) 473.91/297.08 473.91/297.08 We use the processor 'matrix interpretation of dimension 4' to 473.91/297.08 orient following rules strictly. 473.91/297.08 473.91/297.08 Trs: { mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) } 473.91/297.08 473.91/297.08 The induced complexity on above rules (modulo remaining rules) is 473.91/297.08 YES(?,O(n^2)) . These rules are moved into the corresponding weak 473.91/297.08 component(s). 473.91/297.08 473.91/297.08 Sub-proof: 473.91/297.08 ---------- 473.91/297.08 TcT has computed the following triangular matrix interpretation. 473.91/297.08 Note that the diagonal of the component-wise maxima of 473.91/297.08 interpretation-entries contains no more than 2 non-zero entries. 473.91/297.08 473.91/297.08 [1 0 1 0] [0] 473.91/297.08 [active](x1) = [0 0 0 0] x1 + [0] 473.91/297.08 [0 0 0 0] [0] 473.91/297.08 [0 0 0 1] [0] 473.91/297.08 473.91/297.08 [1 0 0 0] [0] 473.91/297.08 [f](x1) = [0 0 0 0] x1 + [0] 473.91/297.08 [0 0 0 0] [1] 473.91/297.08 [0 0 0 1] [1] 473.91/297.08 473.91/297.08 [0] 473.91/297.08 [0] = [0] 473.91/297.08 [0] 473.91/297.08 [0] 473.91/297.08 473.91/297.08 [1 0 0 1] [0] 473.91/297.08 [mark](x1) = [0 0 0 0] x1 + [0] 473.91/297.08 [0 0 0 0] [0] 473.91/297.08 [0 0 0 1] [0] 473.91/297.08 473.91/297.08 [1 0 0 0] [1 0 0 0] [0] 473.91/297.08 [cons](x1, x2) = [0 0 0 0] x1 + [0 0 0 0] x2 + [0] 473.91/297.08 [0 0 0 0] [0 0 0 0] [0] 473.91/297.08 [0 0 0 1] [0 0 0 0] [1] 473.91/297.08 473.91/297.08 [1 0 0 0] [0] 473.91/297.08 [s](x1) = [0 0 0 0] x1 + [0] 473.91/297.08 [0 0 0 0] [0] 473.91/297.08 [0 0 0 1] [0] 473.91/297.08 473.91/297.08 [1 0 0 0] [0] 473.91/297.08 [p](x1) = [0 0 0 0] x1 + [0] 473.91/297.08 [0 0 0 0] [0] 473.91/297.08 [0 0 0 1] [0] 473.91/297.08 473.91/297.08 The order satisfies the following ordering constraints: 473.91/297.08 473.91/297.08 [active(f(0()))] = [1] 473.91/297.08 [0] 473.91/297.08 [0] 473.91/297.08 [1] 473.91/297.08 >= [1] 473.91/297.08 [0] 473.91/297.08 [0] 473.91/297.08 [1] 473.91/297.08 = [mark(cons(0(), f(s(0()))))] 473.91/297.08 473.91/297.08 [active(f(s(0())))] = [1] 473.91/297.08 [0] 473.91/297.08 [0] 473.91/297.08 [1] 473.91/297.08 >= [1] 473.91/297.08 [0] 473.91/297.08 [0] 473.91/297.08 [1] 473.91/297.08 = [mark(f(p(s(0()))))] 473.91/297.08 473.91/297.08 [active(p(s(0())))] = [0] 473.91/297.08 [0] 473.91/297.08 [0] 473.91/297.08 [0] 473.91/297.08 >= [0] 473.91/297.08 [0] 473.91/297.08 [0] 473.91/297.08 [0] 473.91/297.08 = [mark(0())] 473.91/297.08 473.91/297.08 [f(active(X))] = [1 0 1 0] [0] 473.91/297.08 [0 0 0 0] X + [0] 473.91/297.08 [0 0 0 0] [1] 473.91/297.08 [0 0 0 1] [1] 473.91/297.08 >= [1 0 0 0] [0] 473.91/297.08 [0 0 0 0] X + [0] 473.91/297.08 [0 0 0 0] [1] 473.91/297.08 [0 0 0 1] [1] 473.91/297.08 = [f(X)] 473.91/297.08 473.91/297.08 [f(mark(X))] = [1 0 0 1] [0] 473.91/297.08 [0 0 0 0] X + [0] 473.91/297.08 [0 0 0 0] [1] 473.91/297.08 [0 0 0 1] [1] 473.91/297.08 >= [1 0 0 0] [0] 473.91/297.08 [0 0 0 0] X + [0] 473.91/297.08 [0 0 0 0] [1] 473.91/297.08 [0 0 0 1] [1] 473.91/297.08 = [f(X)] 473.91/297.08 473.91/297.09 [mark(f(X))] = [1 0 0 1] [1] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [1] 473.91/297.09 >= [1 0 0 1] [1] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [1] 473.91/297.09 = [active(f(mark(X)))] 473.91/297.09 473.91/297.09 [mark(0())] = [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 >= [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 = [active(0())] 473.91/297.09 473.91/297.09 [mark(cons(X1, X2))] = [1 0 0 1] [1 0 0 0] [1] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0 0 0 0] [1] 473.91/297.09 > [1 0 0 1] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0 0 0 0] [1] 473.91/297.09 = [active(cons(mark(X1), X2))] 473.91/297.09 473.91/297.09 [mark(s(X))] = [1 0 0 1] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 >= [1 0 0 1] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 = [active(s(mark(X)))] 473.91/297.09 473.91/297.09 [mark(p(X))] = [1 0 0 1] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 >= [1 0 0 1] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 = [active(p(mark(X)))] 473.91/297.09 473.91/297.09 [cons(X1, active(X2))] = [1 0 0 0] [1 0 1 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0 0 0 0] [1] 473.91/297.09 >= [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0 0 0 0] [1] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [cons(X1, mark(X2))] = [1 0 0 0] [1 0 0 1] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0 0 0 0] [1] 473.91/297.09 >= [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0 0 0 0] [1] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [cons(active(X1), X2)] = [1 0 1 0] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0 0 0 0] [1] 473.91/297.09 >= [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0 0 0 0] [1] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [cons(mark(X1), X2)] = [1 0 0 1] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0 0 0 0] [1] 473.91/297.09 >= [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0 0 0 0] [1] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [s(active(X))] = [1 0 1 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 = [s(X)] 473.91/297.09 473.91/297.09 [s(mark(X))] = [1 0 0 1] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 = [s(X)] 473.91/297.09 473.91/297.09 [p(active(X))] = [1 0 1 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 = [p(X)] 473.91/297.09 473.91/297.09 [p(mark(X))] = [1 0 0 1] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 = [p(X)] 473.91/297.09 473.91/297.09 473.91/297.09 We return to the main proof. 473.91/297.09 473.91/297.09 We are left with following problem, upon which TcT provides the 473.91/297.09 certificate YES(O(1),O(n^3)). 473.91/297.09 473.91/297.09 Strict Trs: 473.91/297.09 { mark(f(X)) -> active(f(mark(X))) 473.91/297.09 , mark(s(X)) -> active(s(mark(X))) 473.91/297.09 , mark(p(X)) -> active(p(mark(X))) } 473.91/297.09 Weak Trs: 473.91/297.09 { active(f(0())) -> mark(cons(0(), f(s(0())))) 473.91/297.09 , active(f(s(0()))) -> mark(f(p(s(0())))) 473.91/297.09 , active(p(s(0()))) -> mark(0()) 473.91/297.09 , f(active(X)) -> f(X) 473.91/297.09 , f(mark(X)) -> f(X) 473.91/297.09 , mark(0()) -> active(0()) 473.91/297.09 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) 473.91/297.09 , cons(X1, active(X2)) -> cons(X1, X2) 473.91/297.09 , cons(X1, mark(X2)) -> cons(X1, X2) 473.91/297.09 , cons(active(X1), X2) -> cons(X1, X2) 473.91/297.09 , cons(mark(X1), X2) -> cons(X1, X2) 473.91/297.09 , s(active(X)) -> s(X) 473.91/297.09 , s(mark(X)) -> s(X) 473.91/297.09 , p(active(X)) -> p(X) 473.91/297.09 , p(mark(X)) -> p(X) } 473.91/297.09 Obligation: 473.91/297.09 derivational complexity 473.91/297.09 Answer: 473.91/297.09 YES(O(1),O(n^3)) 473.91/297.09 473.91/297.09 We use the processor 'matrix interpretation of dimension 4' to 473.91/297.09 orient following rules strictly. 473.91/297.09 473.91/297.09 Trs: { mark(s(X)) -> active(s(mark(X))) } 473.91/297.09 473.91/297.09 The induced complexity on above rules (modulo remaining rules) is 473.91/297.09 YES(?,O(n^2)) . These rules are moved into the corresponding weak 473.91/297.09 component(s). 473.91/297.09 473.91/297.09 Sub-proof: 473.91/297.09 ---------- 473.91/297.09 TcT has computed the following triangular matrix interpretation. 473.91/297.09 Note that the diagonal of the component-wise maxima of 473.91/297.09 interpretation-entries contains no more than 2 non-zero entries. 473.91/297.09 473.91/297.09 [1 0 1 0] [0] 473.91/297.09 [active](x1) = [0 1 1 0] x1 + [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 473.91/297.09 [1 0 1 1] [0] 473.91/297.09 [f](x1) = [0 1 1 1] x1 + [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 473.91/297.09 [0] 473.91/297.09 [0] = [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 473.91/297.09 [1 1 1 1] [0] 473.91/297.09 [mark](x1) = [0 1 1 0] x1 + [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 473.91/297.09 [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [cons](x1, x2) = [0 1 1 1] x1 + [0 0 0 0] x2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 473.91/297.09 [1 0 0 0] [0] 473.91/297.09 [s](x1) = [0 1 1 1] x1 + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 473.91/297.09 [1 0 0 0] [0] 473.91/297.09 [p](x1) = [0 1 1 1] x1 + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 473.91/297.09 The order satisfies the following ordering constraints: 473.91/297.09 473.91/297.09 [active(f(0()))] = [1] 473.91/297.09 [1] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 >= [1] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 = [mark(cons(0(), f(s(0()))))] 473.91/297.09 473.91/297.09 [active(f(s(0())))] = [2] 473.91/297.09 [2] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 >= [2] 473.91/297.09 [2] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 = [mark(f(p(s(0()))))] 473.91/297.09 473.91/297.09 [active(p(s(0())))] = [0] 473.91/297.09 [1] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 >= [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 = [mark(0())] 473.91/297.09 473.91/297.09 [f(active(X))] = [1 0 1 1] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 >= [1 0 1 1] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 = [f(X)] 473.91/297.09 473.91/297.09 [f(mark(X))] = [1 1 1 2] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 >= [1 0 1 1] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 = [f(X)] 473.91/297.09 473.91/297.09 [mark(f(X))] = [1 1 2 2] [1] 473.91/297.09 [0 1 1 1] X + [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 >= [1 1 1 2] [1] 473.91/297.09 [0 1 1 1] X + [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 = [active(f(mark(X)))] 473.91/297.09 473.91/297.09 [mark(0())] = [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 >= [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 = [active(0())] 473.91/297.09 473.91/297.09 [mark(cons(X1, X2))] = [1 1 1 1] [1 0 0 0] [0] 473.91/297.09 [0 1 1 1] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 >= [1 1 1 1] [1 0 0 0] [0] 473.91/297.09 [0 1 1 1] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 = [active(cons(mark(X1), X2))] 473.91/297.09 473.91/297.09 [mark(s(X))] = [1 1 1 1] [1] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 > [1 1 1 1] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 = [active(s(mark(X)))] 473.91/297.09 473.91/297.09 [mark(p(X))] = [1 1 1 1] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 >= [1 1 1 1] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 = [active(p(mark(X)))] 473.91/297.09 473.91/297.09 [cons(X1, active(X2))] = [1 0 0 0] [1 0 1 0] [0] 473.91/297.09 [0 1 1 1] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [0 1 1 1] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [cons(X1, mark(X2))] = [1 0 0 0] [1 1 1 1] [0] 473.91/297.09 [0 1 1 1] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [0 1 1 1] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [cons(active(X1), X2)] = [1 0 1 0] [1 0 0 0] [0] 473.91/297.09 [0 1 1 1] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [0 1 1 1] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [cons(mark(X1), X2)] = [1 1 1 1] [1 0 0 0] [0] 473.91/297.09 [0 1 1 1] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [0 1 1 1] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [s(active(X))] = [1 0 1 0] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 = [s(X)] 473.91/297.09 473.91/297.09 [s(mark(X))] = [1 1 1 1] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 = [s(X)] 473.91/297.09 473.91/297.09 [p(active(X))] = [1 0 1 0] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 = [p(X)] 473.91/297.09 473.91/297.09 [p(mark(X))] = [1 1 1 1] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 1 1 1] X + [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 = [p(X)] 473.91/297.09 473.91/297.09 473.91/297.09 We return to the main proof. 473.91/297.09 473.91/297.09 We are left with following problem, upon which TcT provides the 473.91/297.09 certificate YES(O(1),O(n^3)). 473.91/297.09 473.91/297.09 Strict Trs: 473.91/297.09 { mark(f(X)) -> active(f(mark(X))) 473.91/297.09 , mark(p(X)) -> active(p(mark(X))) } 473.91/297.09 Weak Trs: 473.91/297.09 { active(f(0())) -> mark(cons(0(), f(s(0())))) 473.91/297.09 , active(f(s(0()))) -> mark(f(p(s(0())))) 473.91/297.09 , active(p(s(0()))) -> mark(0()) 473.91/297.09 , f(active(X)) -> f(X) 473.91/297.09 , f(mark(X)) -> f(X) 473.91/297.09 , mark(0()) -> active(0()) 473.91/297.09 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) 473.91/297.09 , mark(s(X)) -> active(s(mark(X))) 473.91/297.09 , cons(X1, active(X2)) -> cons(X1, X2) 473.91/297.09 , cons(X1, mark(X2)) -> cons(X1, X2) 473.91/297.09 , cons(active(X1), X2) -> cons(X1, X2) 473.91/297.09 , cons(mark(X1), X2) -> cons(X1, X2) 473.91/297.09 , s(active(X)) -> s(X) 473.91/297.09 , s(mark(X)) -> s(X) 473.91/297.09 , p(active(X)) -> p(X) 473.91/297.09 , p(mark(X)) -> p(X) } 473.91/297.09 Obligation: 473.91/297.09 derivational complexity 473.91/297.09 Answer: 473.91/297.09 YES(O(1),O(n^3)) 473.91/297.09 473.91/297.09 We use the processor 'matrix interpretation of dimension 4' to 473.91/297.09 orient following rules strictly. 473.91/297.09 473.91/297.09 Trs: { mark(p(X)) -> active(p(mark(X))) } 473.91/297.09 473.91/297.09 The induced complexity on above rules (modulo remaining rules) is 473.91/297.09 YES(?,O(n^3)) . These rules are moved into the corresponding weak 473.91/297.09 component(s). 473.91/297.09 473.91/297.09 Sub-proof: 473.91/297.09 ---------- 473.91/297.09 TcT has computed the following triangular matrix interpretation. 473.91/297.09 Note that the diagonal of the component-wise maxima of 473.91/297.09 interpretation-entries contains no more than 3 non-zero entries. 473.91/297.09 473.91/297.09 [1 1 0 0] [0] 473.91/297.09 [active](x1) = [0 0 0 0] x1 + [0] 473.91/297.09 [0 0 1 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 473.91/297.09 [1 0 0 0] [0] 473.91/297.09 [f](x1) = [0 0 0 1] x1 + [1] 473.91/297.09 [0 0 1 1] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 473.91/297.09 [0] 473.91/297.09 [0] = [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 473.91/297.09 [1 0 1 0] [0] 473.91/297.09 [mark](x1) = [0 0 0 0] x1 + [0] 473.91/297.09 [0 0 1 0] [0] 473.91/297.09 [0 0 0 1] [0] 473.91/297.09 473.91/297.09 [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [cons](x1, x2) = [0 0 0 0] x1 + [0 0 0 0] x2 + [0] 473.91/297.09 [0 0 1 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 473.91/297.09 [1 0 0 0] [0] 473.91/297.09 [s](x1) = [0 0 0 0] x1 + [0] 473.91/297.09 [0 0 1 0] [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 473.91/297.09 [1 0 0 0] [0] 473.91/297.09 [p](x1) = [0 0 0 0] x1 + [0] 473.91/297.09 [0 0 1 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 473.91/297.09 The order satisfies the following ordering constraints: 473.91/297.09 473.91/297.09 [active(f(0()))] = [1] 473.91/297.09 [0] 473.91/297.09 [1] 473.91/297.09 [0] 473.91/297.09 > [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 = [mark(cons(0(), f(s(0()))))] 473.91/297.09 473.91/297.09 [active(f(s(0())))] = [2] 473.91/297.09 [0] 473.91/297.09 [2] 473.91/297.09 [0] 473.91/297.09 >= [2] 473.91/297.09 [0] 473.91/297.09 [2] 473.91/297.09 [0] 473.91/297.09 = [mark(f(p(s(0()))))] 473.91/297.09 473.91/297.09 [active(p(s(0())))] = [0] 473.91/297.09 [0] 473.91/297.09 [1] 473.91/297.09 [0] 473.91/297.09 >= [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 = [mark(0())] 473.91/297.09 473.91/297.09 [f(active(X))] = [1 1 0 0] [0] 473.91/297.09 [0 0 0 1] X + [1] 473.91/297.09 [0 0 1 1] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 0 0 1] X + [1] 473.91/297.09 [0 0 1 1] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 = [f(X)] 473.91/297.09 473.91/297.09 [f(mark(X))] = [1 0 1 0] [0] 473.91/297.09 [0 0 0 1] X + [1] 473.91/297.09 [0 0 1 1] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 0 0 1] X + [1] 473.91/297.09 [0 0 1 1] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 = [f(X)] 473.91/297.09 473.91/297.09 [mark(f(X))] = [1 0 1 1] [1] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 1] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 >= [1 0 1 1] [1] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 1] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 = [active(f(mark(X)))] 473.91/297.09 473.91/297.09 [mark(0())] = [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 >= [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 = [active(0())] 473.91/297.09 473.91/297.09 [mark(cons(X1, X2))] = [1 0 1 0] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 1 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 >= [1 0 1 0] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 1 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 = [active(cons(mark(X1), X2))] 473.91/297.09 473.91/297.09 [mark(s(X))] = [1 0 1 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 0] [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 >= [1 0 1 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 0] [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 = [active(s(mark(X)))] 473.91/297.09 473.91/297.09 [mark(p(X))] = [1 0 1 0] [1] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 > [1 0 1 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 = [active(p(mark(X)))] 473.91/297.09 473.91/297.09 [cons(X1, active(X2))] = [1 0 0 0] [1 1 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 1 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 1 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [cons(X1, mark(X2))] = [1 0 0 0] [1 0 1 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 1 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 1 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [cons(active(X1), X2)] = [1 1 0 0] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 1 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 1 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [cons(mark(X1), X2)] = [1 0 1 0] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 1 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X1 + [0 0 0 0] X2 + [0] 473.91/297.09 [0 0 1 0] [0 0 0 0] [0] 473.91/297.09 [0 0 0 0] [0 0 0 0] [0] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [s(active(X))] = [1 1 0 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 0] [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 0] [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 = [s(X)] 473.91/297.09 473.91/297.09 [s(mark(X))] = [1 0 1 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 0] [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 0] [0] 473.91/297.09 [0 0 0 0] [1] 473.91/297.09 = [s(X)] 473.91/297.09 473.91/297.09 [p(active(X))] = [1 1 0 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 = [p(X)] 473.91/297.09 473.91/297.09 [p(mark(X))] = [1 0 1 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 >= [1 0 0 0] [0] 473.91/297.09 [0 0 0 0] X + [0] 473.91/297.09 [0 0 1 0] [1] 473.91/297.09 [0 0 0 0] [0] 473.91/297.09 = [p(X)] 473.91/297.09 473.91/297.09 473.91/297.09 We return to the main proof. 473.91/297.09 473.91/297.09 We are left with following problem, upon which TcT provides the 473.91/297.09 certificate YES(O(1),O(n^3)). 473.91/297.09 473.91/297.09 Strict Trs: { mark(f(X)) -> active(f(mark(X))) } 473.91/297.09 Weak Trs: 473.91/297.09 { active(f(0())) -> mark(cons(0(), f(s(0())))) 473.91/297.09 , active(f(s(0()))) -> mark(f(p(s(0())))) 473.91/297.09 , active(p(s(0()))) -> mark(0()) 473.91/297.09 , f(active(X)) -> f(X) 473.91/297.09 , f(mark(X)) -> f(X) 473.91/297.09 , mark(0()) -> active(0()) 473.91/297.09 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) 473.91/297.09 , mark(s(X)) -> active(s(mark(X))) 473.91/297.09 , mark(p(X)) -> active(p(mark(X))) 473.91/297.09 , cons(X1, active(X2)) -> cons(X1, X2) 473.91/297.09 , cons(X1, mark(X2)) -> cons(X1, X2) 473.91/297.09 , cons(active(X1), X2) -> cons(X1, X2) 473.91/297.09 , cons(mark(X1), X2) -> cons(X1, X2) 473.91/297.09 , s(active(X)) -> s(X) 473.91/297.09 , s(mark(X)) -> s(X) 473.91/297.09 , p(active(X)) -> p(X) 473.91/297.09 , p(mark(X)) -> p(X) } 473.91/297.09 Obligation: 473.91/297.09 derivational complexity 473.91/297.09 Answer: 473.91/297.09 YES(O(1),O(n^3)) 473.91/297.09 473.91/297.09 We use the processor 'matrix interpretation of dimension 3' to 473.91/297.09 orient following rules strictly. 473.91/297.09 473.91/297.09 Trs: { mark(f(X)) -> active(f(mark(X))) } 473.91/297.09 473.91/297.09 The induced complexity on above rules (modulo remaining rules) is 473.91/297.09 YES(?,O(n^3)) . These rules are moved into the corresponding weak 473.91/297.09 component(s). 473.91/297.09 473.91/297.09 Sub-proof: 473.91/297.09 ---------- 473.91/297.09 TcT has computed the following triangular matrix interpretation. 473.91/297.09 473.91/297.09 [1 0 1] [0] 473.91/297.09 [active](x1) = [0 1 0] x1 + [0] 473.91/297.09 [0 0 1] [0] 473.91/297.09 473.91/297.09 [1 0 1] [0] 473.91/297.09 [f](x1) = [0 1 1] x1 + [2] 473.91/297.09 [0 0 1] [2] 473.91/297.09 473.91/297.09 [0] 473.91/297.09 [0] = [0] 473.91/297.09 [0] 473.91/297.09 473.91/297.09 [1 1 1] [0] 473.91/297.09 [mark](x1) = [0 1 0] x1 + [0] 473.91/297.09 [0 0 1] [0] 473.91/297.09 473.91/297.09 [1 0 0] [1 0 0] [0] 473.91/297.09 [cons](x1, x2) = [0 1 1] x1 + [0 0 0] x2 + [0] 473.91/297.09 [0 0 0] [0 0 0] [0] 473.91/297.09 473.91/297.09 [1 0 0] [0] 473.91/297.09 [s](x1) = [0 1 2] x1 + [0] 473.91/297.09 [0 0 0] [2] 473.91/297.09 473.91/297.09 [1 0 0] [0] 473.91/297.09 [p](x1) = [0 1 1] x1 + [0] 473.91/297.09 [0 0 0] [0] 473.91/297.09 473.91/297.09 The order satisfies the following ordering constraints: 473.91/297.09 473.91/297.09 [active(f(0()))] = [2] 473.91/297.09 [2] 473.91/297.09 [2] 473.91/297.09 >= [2] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 = [mark(cons(0(), f(s(0()))))] 473.91/297.09 473.91/297.09 [active(f(s(0())))] = [6] 473.91/297.09 [4] 473.91/297.09 [4] 473.91/297.09 >= [6] 473.91/297.09 [4] 473.91/297.09 [2] 473.91/297.09 = [mark(f(p(s(0()))))] 473.91/297.09 473.91/297.09 [active(p(s(0())))] = [0] 473.91/297.09 [2] 473.91/297.09 [0] 473.91/297.09 >= [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 = [mark(0())] 473.91/297.09 473.91/297.09 [f(active(X))] = [1 0 2] [0] 473.91/297.09 [0 1 1] X + [2] 473.91/297.09 [0 0 1] [2] 473.91/297.09 >= [1 0 1] [0] 473.91/297.09 [0 1 1] X + [2] 473.91/297.09 [0 0 1] [2] 473.91/297.09 = [f(X)] 473.91/297.09 473.91/297.09 [f(mark(X))] = [1 1 2] [0] 473.91/297.09 [0 1 1] X + [2] 473.91/297.09 [0 0 1] [2] 473.91/297.09 >= [1 0 1] [0] 473.91/297.09 [0 1 1] X + [2] 473.91/297.09 [0 0 1] [2] 473.91/297.09 = [f(X)] 473.91/297.09 473.91/297.09 [mark(f(X))] = [1 1 3] [4] 473.91/297.09 [0 1 1] X + [2] 473.91/297.09 [0 0 1] [2] 473.91/297.09 > [1 1 3] [2] 473.91/297.09 [0 1 1] X + [2] 473.91/297.09 [0 0 1] [2] 473.91/297.09 = [active(f(mark(X)))] 473.91/297.09 473.91/297.09 [mark(0())] = [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 >= [0] 473.91/297.09 [0] 473.91/297.09 [0] 473.91/297.09 = [active(0())] 473.91/297.09 473.91/297.09 [mark(cons(X1, X2))] = [1 1 1] [1 0 0] [0] 473.91/297.09 [0 1 1] X1 + [0 0 0] X2 + [0] 473.91/297.09 [0 0 0] [0 0 0] [0] 473.91/297.09 >= [1 1 1] [1 0 0] [0] 473.91/297.09 [0 1 1] X1 + [0 0 0] X2 + [0] 473.91/297.09 [0 0 0] [0 0 0] [0] 473.91/297.09 = [active(cons(mark(X1), X2))] 473.91/297.09 473.91/297.09 [mark(s(X))] = [1 1 2] [2] 473.91/297.09 [0 1 2] X + [0] 473.91/297.09 [0 0 0] [2] 473.91/297.09 >= [1 1 1] [2] 473.91/297.09 [0 1 2] X + [0] 473.91/297.09 [0 0 0] [2] 473.91/297.09 = [active(s(mark(X)))] 473.91/297.09 473.91/297.09 [mark(p(X))] = [1 1 1] [0] 473.91/297.09 [0 1 1] X + [0] 473.91/297.09 [0 0 0] [0] 473.91/297.09 >= [1 1 1] [0] 473.91/297.09 [0 1 1] X + [0] 473.91/297.09 [0 0 0] [0] 473.91/297.09 = [active(p(mark(X)))] 473.91/297.09 473.91/297.09 [cons(X1, active(X2))] = [1 0 0] [1 0 1] [0] 473.91/297.09 [0 1 1] X1 + [0 0 0] X2 + [0] 473.91/297.09 [0 0 0] [0 0 0] [0] 473.91/297.09 >= [1 0 0] [1 0 0] [0] 473.91/297.09 [0 1 1] X1 + [0 0 0] X2 + [0] 473.91/297.09 [0 0 0] [0 0 0] [0] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [cons(X1, mark(X2))] = [1 0 0] [1 1 1] [0] 473.91/297.09 [0 1 1] X1 + [0 0 0] X2 + [0] 473.91/297.09 [0 0 0] [0 0 0] [0] 473.91/297.09 >= [1 0 0] [1 0 0] [0] 473.91/297.09 [0 1 1] X1 + [0 0 0] X2 + [0] 473.91/297.09 [0 0 0] [0 0 0] [0] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [cons(active(X1), X2)] = [1 0 1] [1 0 0] [0] 473.91/297.09 [0 1 1] X1 + [0 0 0] X2 + [0] 473.91/297.09 [0 0 0] [0 0 0] [0] 473.91/297.09 >= [1 0 0] [1 0 0] [0] 473.91/297.09 [0 1 1] X1 + [0 0 0] X2 + [0] 473.91/297.09 [0 0 0] [0 0 0] [0] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [cons(mark(X1), X2)] = [1 1 1] [1 0 0] [0] 473.91/297.09 [0 1 1] X1 + [0 0 0] X2 + [0] 473.91/297.09 [0 0 0] [0 0 0] [0] 473.91/297.09 >= [1 0 0] [1 0 0] [0] 473.91/297.09 [0 1 1] X1 + [0 0 0] X2 + [0] 473.91/297.09 [0 0 0] [0 0 0] [0] 473.91/297.09 = [cons(X1, X2)] 473.91/297.09 473.91/297.09 [s(active(X))] = [1 0 1] [0] 473.91/297.09 [0 1 2] X + [0] 473.91/297.09 [0 0 0] [2] 473.91/297.09 >= [1 0 0] [0] 473.91/297.09 [0 1 2] X + [0] 473.91/297.09 [0 0 0] [2] 473.91/297.09 = [s(X)] 473.91/297.09 473.91/297.09 [s(mark(X))] = [1 1 1] [0] 473.91/297.09 [0 1 2] X + [0] 473.91/297.09 [0 0 0] [2] 473.91/297.09 >= [1 0 0] [0] 473.91/297.09 [0 1 2] X + [0] 473.91/297.09 [0 0 0] [2] 473.91/297.09 = [s(X)] 473.91/297.09 473.91/297.09 [p(active(X))] = [1 0 1] [0] 473.91/297.09 [0 1 1] X + [0] 473.91/297.09 [0 0 0] [0] 473.91/297.09 >= [1 0 0] [0] 473.91/297.09 [0 1 1] X + [0] 473.91/297.09 [0 0 0] [0] 473.91/297.09 = [p(X)] 473.91/297.09 473.91/297.09 [p(mark(X))] = [1 1 1] [0] 473.91/297.09 [0 1 1] X + [0] 473.91/297.09 [0 0 0] [0] 473.91/297.09 >= [1 0 0] [0] 473.91/297.09 [0 1 1] X + [0] 473.91/297.09 [0 0 0] [0] 473.91/297.09 = [p(X)] 473.91/297.09 473.91/297.09 473.91/297.09 We return to the main proof. 473.91/297.09 473.91/297.09 We are left with following problem, upon which TcT provides the 473.91/297.09 certificate YES(O(1),O(1)). 473.91/297.09 473.91/297.09 Weak Trs: 473.91/297.09 { active(f(0())) -> mark(cons(0(), f(s(0())))) 473.91/297.09 , active(f(s(0()))) -> mark(f(p(s(0())))) 473.91/297.09 , active(p(s(0()))) -> mark(0()) 473.91/297.09 , f(active(X)) -> f(X) 473.91/297.09 , f(mark(X)) -> f(X) 473.91/297.09 , mark(f(X)) -> active(f(mark(X))) 473.91/297.09 , mark(0()) -> active(0()) 473.91/297.09 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) 473.91/297.09 , mark(s(X)) -> active(s(mark(X))) 473.91/297.09 , mark(p(X)) -> active(p(mark(X))) 473.91/297.09 , cons(X1, active(X2)) -> cons(X1, X2) 473.91/297.09 , cons(X1, mark(X2)) -> cons(X1, X2) 473.91/297.09 , cons(active(X1), X2) -> cons(X1, X2) 473.91/297.09 , cons(mark(X1), X2) -> cons(X1, X2) 473.91/297.09 , s(active(X)) -> s(X) 473.91/297.09 , s(mark(X)) -> s(X) 473.91/297.09 , p(active(X)) -> p(X) 473.91/297.09 , p(mark(X)) -> p(X) } 473.91/297.09 Obligation: 473.91/297.09 derivational complexity 473.91/297.09 Answer: 473.91/297.09 YES(O(1),O(1)) 473.91/297.09 473.91/297.09 Empty rules are trivially bounded 473.91/297.09 473.91/297.09 Hurray, we answered YES(O(1),O(n^3)) 474.05/297.12 EOF