YES(O(1),O(n^3)) 213.53/60.04 YES(O(1),O(n^3)) 213.53/60.04 213.53/60.04 We are left with following problem, upon which TcT provides the 213.53/60.04 certificate YES(O(1),O(n^3)). 213.53/60.04 213.53/60.04 Strict Trs: 213.53/60.04 { active(zeros()) -> mark(cons(0(), zeros())) 213.53/60.04 , active(tail(cons(X, XS))) -> mark(XS) 213.53/60.04 , mark(zeros()) -> active(zeros()) 213.53/60.04 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) 213.53/60.04 , mark(0()) -> active(0()) 213.53/60.04 , mark(tail(X)) -> active(tail(mark(X))) 213.53/60.04 , cons(X1, active(X2)) -> cons(X1, X2) 213.53/60.04 , cons(X1, mark(X2)) -> cons(X1, X2) 213.53/60.04 , cons(active(X1), X2) -> cons(X1, X2) 213.53/60.04 , cons(mark(X1), X2) -> cons(X1, X2) 213.53/60.04 , tail(active(X)) -> tail(X) 213.53/60.04 , tail(mark(X)) -> tail(X) } 213.53/60.04 Obligation: 213.53/60.04 derivational complexity 213.53/60.04 Answer: 213.53/60.04 YES(O(1),O(n^3)) 213.53/60.04 213.53/60.04 The weightgap principle applies (using the following nonconstant 213.53/60.04 growth matrix-interpretation) 213.53/60.04 213.53/60.04 TcT has computed the following triangular matrix interpretation. 213.53/60.04 Note that the diagonal of the component-wise maxima of 213.53/60.04 interpretation-entries contains no more than 1 non-zero entries. 213.53/60.04 213.53/60.04 [active](x1) = [1] x1 + [0] 213.53/60.04 213.53/60.04 [zeros] = [0] 213.53/60.04 213.53/60.04 [mark](x1) = [1] x1 + [1] 213.53/60.04 213.53/60.04 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 213.53/60.04 213.53/60.04 [0] = [0] 213.53/60.04 213.53/60.04 [tail](x1) = [1] x1 + [0] 213.53/60.04 213.53/60.04 The order satisfies the following ordering constraints: 213.53/60.04 213.53/60.04 [active(zeros())] = [0] 213.53/60.04 ? [1] 213.53/60.04 = [mark(cons(0(), zeros()))] 213.53/60.04 213.53/60.04 [active(tail(cons(X, XS)))] = [1] X + [1] XS + [0] 213.53/60.04 ? [1] XS + [1] 213.53/60.04 = [mark(XS)] 213.53/60.04 213.53/60.04 [mark(zeros())] = [1] 213.53/60.04 > [0] 213.53/60.04 = [active(zeros())] 213.53/60.04 213.53/60.04 [mark(cons(X1, X2))] = [1] X1 + [1] X2 + [1] 213.53/60.04 >= [1] X1 + [1] X2 + [1] 213.53/60.04 = [active(cons(mark(X1), X2))] 213.53/60.04 213.53/60.04 [mark(0())] = [1] 213.53/60.04 > [0] 213.53/60.04 = [active(0())] 213.53/60.04 213.53/60.04 [mark(tail(X))] = [1] X + [1] 213.53/60.04 >= [1] X + [1] 213.53/60.04 = [active(tail(mark(X)))] 213.53/60.04 213.53/60.04 [cons(X1, active(X2))] = [1] X1 + [1] X2 + [0] 213.53/60.04 >= [1] X1 + [1] X2 + [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(X1, mark(X2))] = [1] X1 + [1] X2 + [1] 213.53/60.04 > [1] X1 + [1] X2 + [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(active(X1), X2)] = [1] X1 + [1] X2 + [0] 213.53/60.04 >= [1] X1 + [1] X2 + [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(mark(X1), X2)] = [1] X1 + [1] X2 + [1] 213.53/60.04 > [1] X1 + [1] X2 + [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [tail(active(X))] = [1] X + [0] 213.53/60.04 >= [1] X + [0] 213.53/60.04 = [tail(X)] 213.53/60.04 213.53/60.04 [tail(mark(X))] = [1] X + [1] 213.53/60.04 > [1] X + [0] 213.53/60.04 = [tail(X)] 213.53/60.04 213.53/60.04 213.53/60.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 213.53/60.04 213.53/60.04 We are left with following problem, upon which TcT provides the 213.53/60.04 certificate YES(O(1),O(n^3)). 213.53/60.04 213.53/60.04 Strict Trs: 213.53/60.04 { active(zeros()) -> mark(cons(0(), zeros())) 213.53/60.04 , active(tail(cons(X, XS))) -> mark(XS) 213.53/60.04 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) 213.53/60.04 , mark(tail(X)) -> active(tail(mark(X))) 213.53/60.04 , cons(X1, active(X2)) -> cons(X1, X2) 213.53/60.04 , cons(active(X1), X2) -> cons(X1, X2) 213.53/60.04 , tail(active(X)) -> tail(X) } 213.53/60.04 Weak Trs: 213.53/60.04 { mark(zeros()) -> active(zeros()) 213.53/60.04 , mark(0()) -> active(0()) 213.53/60.04 , cons(X1, mark(X2)) -> cons(X1, X2) 213.53/60.04 , cons(mark(X1), X2) -> cons(X1, X2) 213.53/60.04 , tail(mark(X)) -> tail(X) } 213.53/60.04 Obligation: 213.53/60.04 derivational complexity 213.53/60.04 Answer: 213.53/60.04 YES(O(1),O(n^3)) 213.53/60.04 213.53/60.04 We use the processor 'matrix interpretation of dimension 1' to 213.53/60.04 orient following rules strictly. 213.53/60.04 213.53/60.04 Trs: { active(tail(cons(X, XS))) -> mark(XS) } 213.53/60.04 213.53/60.04 The induced complexity on above rules (modulo remaining rules) is 213.53/60.04 YES(?,O(n^1)) . These rules are moved into the corresponding weak 213.53/60.04 component(s). 213.53/60.04 213.53/60.04 Sub-proof: 213.53/60.04 ---------- 213.53/60.04 TcT has computed the following triangular matrix interpretation. 213.53/60.04 213.53/60.04 [active](x1) = [1] x1 + [0] 213.53/60.04 213.53/60.04 [zeros] = [0] 213.53/60.04 213.53/60.04 [mark](x1) = [1] x1 + [0] 213.53/60.04 213.53/60.04 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 213.53/60.04 213.53/60.04 [0] = [0] 213.53/60.04 213.53/60.04 [tail](x1) = [1] x1 + [1] 213.53/60.04 213.53/60.04 The order satisfies the following ordering constraints: 213.53/60.04 213.53/60.04 [active(zeros())] = [0] 213.53/60.04 >= [0] 213.53/60.04 = [mark(cons(0(), zeros()))] 213.53/60.04 213.53/60.04 [active(tail(cons(X, XS)))] = [1] X + [1] XS + [1] 213.53/60.04 > [1] XS + [0] 213.53/60.04 = [mark(XS)] 213.53/60.04 213.53/60.04 [mark(zeros())] = [0] 213.53/60.04 >= [0] 213.53/60.04 = [active(zeros())] 213.53/60.04 213.53/60.04 [mark(cons(X1, X2))] = [1] X1 + [1] X2 + [0] 213.53/60.04 >= [1] X1 + [1] X2 + [0] 213.53/60.04 = [active(cons(mark(X1), X2))] 213.53/60.04 213.53/60.04 [mark(0())] = [0] 213.53/60.04 >= [0] 213.53/60.04 = [active(0())] 213.53/60.04 213.53/60.04 [mark(tail(X))] = [1] X + [1] 213.53/60.04 >= [1] X + [1] 213.53/60.04 = [active(tail(mark(X)))] 213.53/60.04 213.53/60.04 [cons(X1, active(X2))] = [1] X1 + [1] X2 + [0] 213.53/60.04 >= [1] X1 + [1] X2 + [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(X1, mark(X2))] = [1] X1 + [1] X2 + [0] 213.53/60.04 >= [1] X1 + [1] X2 + [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(active(X1), X2)] = [1] X1 + [1] X2 + [0] 213.53/60.04 >= [1] X1 + [1] X2 + [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(mark(X1), X2)] = [1] X1 + [1] X2 + [0] 213.53/60.04 >= [1] X1 + [1] X2 + [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [tail(active(X))] = [1] X + [1] 213.53/60.04 >= [1] X + [1] 213.53/60.04 = [tail(X)] 213.53/60.04 213.53/60.04 [tail(mark(X))] = [1] X + [1] 213.53/60.04 >= [1] X + [1] 213.53/60.04 = [tail(X)] 213.53/60.04 213.53/60.04 213.53/60.04 We return to the main proof. 213.53/60.04 213.53/60.04 We are left with following problem, upon which TcT provides the 213.53/60.04 certificate YES(O(1),O(n^3)). 213.53/60.04 213.53/60.04 Strict Trs: 213.53/60.04 { active(zeros()) -> mark(cons(0(), zeros())) 213.53/60.04 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) 213.53/60.04 , mark(tail(X)) -> active(tail(mark(X))) 213.53/60.04 , cons(X1, active(X2)) -> cons(X1, X2) 213.53/60.04 , cons(active(X1), X2) -> cons(X1, X2) 213.53/60.04 , tail(active(X)) -> tail(X) } 213.53/60.04 Weak Trs: 213.53/60.04 { active(tail(cons(X, XS))) -> mark(XS) 213.53/60.04 , mark(zeros()) -> active(zeros()) 213.53/60.04 , mark(0()) -> active(0()) 213.53/60.04 , cons(X1, mark(X2)) -> cons(X1, X2) 213.53/60.04 , cons(mark(X1), X2) -> cons(X1, X2) 213.53/60.04 , tail(mark(X)) -> tail(X) } 213.53/60.04 Obligation: 213.53/60.04 derivational complexity 213.53/60.04 Answer: 213.53/60.04 YES(O(1),O(n^3)) 213.53/60.04 213.53/60.04 The weightgap principle applies (using the following nonconstant 213.53/60.04 growth matrix-interpretation) 213.53/60.04 213.53/60.04 TcT has computed the following triangular matrix interpretation. 213.53/60.04 Note that the diagonal of the component-wise maxima of 213.53/60.04 interpretation-entries contains no more than 1 non-zero entries. 213.53/60.04 213.53/60.04 [active](x1) = [1] x1 + [1] 213.53/60.04 213.53/60.04 [zeros] = [0] 213.53/60.04 213.53/60.04 [mark](x1) = [1] x1 + [2] 213.53/60.04 213.53/60.04 [cons](x1, x2) = [1] x1 + [1] x2 + [2] 213.53/60.04 213.53/60.04 [0] = [0] 213.53/60.04 213.53/60.04 [tail](x1) = [1] x1 + [0] 213.53/60.04 213.53/60.04 The order satisfies the following ordering constraints: 213.53/60.04 213.53/60.04 [active(zeros())] = [1] 213.53/60.04 ? [4] 213.53/60.04 = [mark(cons(0(), zeros()))] 213.53/60.04 213.53/60.04 [active(tail(cons(X, XS)))] = [1] X + [1] XS + [3] 213.53/60.04 > [1] XS + [2] 213.53/60.04 = [mark(XS)] 213.53/60.04 213.53/60.04 [mark(zeros())] = [2] 213.53/60.04 > [1] 213.53/60.04 = [active(zeros())] 213.53/60.04 213.53/60.04 [mark(cons(X1, X2))] = [1] X1 + [1] X2 + [4] 213.53/60.04 ? [1] X1 + [1] X2 + [5] 213.53/60.04 = [active(cons(mark(X1), X2))] 213.53/60.04 213.53/60.04 [mark(0())] = [2] 213.53/60.04 > [1] 213.53/60.04 = [active(0())] 213.53/60.04 213.53/60.04 [mark(tail(X))] = [1] X + [2] 213.53/60.04 ? [1] X + [3] 213.53/60.04 = [active(tail(mark(X)))] 213.53/60.04 213.53/60.04 [cons(X1, active(X2))] = [1] X1 + [1] X2 + [3] 213.53/60.04 > [1] X1 + [1] X2 + [2] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(X1, mark(X2))] = [1] X1 + [1] X2 + [4] 213.53/60.04 > [1] X1 + [1] X2 + [2] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(active(X1), X2)] = [1] X1 + [1] X2 + [3] 213.53/60.04 > [1] X1 + [1] X2 + [2] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(mark(X1), X2)] = [1] X1 + [1] X2 + [4] 213.53/60.04 > [1] X1 + [1] X2 + [2] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [tail(active(X))] = [1] X + [1] 213.53/60.04 > [1] X + [0] 213.53/60.04 = [tail(X)] 213.53/60.04 213.53/60.04 [tail(mark(X))] = [1] X + [2] 213.53/60.04 > [1] X + [0] 213.53/60.04 = [tail(X)] 213.53/60.04 213.53/60.04 213.53/60.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 213.53/60.04 213.53/60.04 We are left with following problem, upon which TcT provides the 213.53/60.04 certificate YES(O(1),O(n^3)). 213.53/60.04 213.53/60.04 Strict Trs: 213.53/60.04 { active(zeros()) -> mark(cons(0(), zeros())) 213.53/60.04 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) 213.53/60.04 , mark(tail(X)) -> active(tail(mark(X))) } 213.53/60.04 Weak Trs: 213.53/60.04 { active(tail(cons(X, XS))) -> mark(XS) 213.53/60.04 , mark(zeros()) -> active(zeros()) 213.53/60.04 , mark(0()) -> active(0()) 213.53/60.04 , cons(X1, active(X2)) -> cons(X1, X2) 213.53/60.04 , cons(X1, mark(X2)) -> cons(X1, X2) 213.53/60.04 , cons(active(X1), X2) -> cons(X1, X2) 213.53/60.04 , cons(mark(X1), X2) -> cons(X1, X2) 213.53/60.04 , tail(active(X)) -> tail(X) 213.53/60.04 , tail(mark(X)) -> tail(X) } 213.53/60.04 Obligation: 213.53/60.04 derivational complexity 213.53/60.04 Answer: 213.53/60.04 YES(O(1),O(n^3)) 213.53/60.04 213.53/60.04 We use the processor 'matrix interpretation of dimension 2' to 213.53/60.04 orient following rules strictly. 213.53/60.04 213.53/60.04 Trs: { mark(tail(X)) -> active(tail(mark(X))) } 213.53/60.04 213.53/60.04 The induced complexity on above rules (modulo remaining rules) is 213.53/60.04 YES(?,O(n^2)) . These rules are moved into the corresponding weak 213.53/60.04 component(s). 213.53/60.04 213.53/60.04 Sub-proof: 213.53/60.04 ---------- 213.53/60.04 TcT has computed the following triangular matrix interpretation. 213.53/60.04 213.53/60.04 [active](x1) = [1 0] x1 + [0] 213.53/60.04 [0 1] [0] 213.53/60.04 213.53/60.04 [zeros] = [0] 213.53/60.04 [0] 213.53/60.04 213.53/60.04 [mark](x1) = [1 1] x1 + [0] 213.53/60.04 [0 1] [0] 213.53/60.04 213.53/60.04 [cons](x1, x2) = [1 0] x1 + [1 1] x2 + [0] 213.53/60.04 [0 1] [0 1] [0] 213.53/60.04 213.53/60.04 [0] = [0] 213.53/60.04 [0] 213.53/60.04 213.53/60.04 [tail](x1) = [1 0] x1 + [0] 213.53/60.04 [0 1] [1] 213.53/60.04 213.53/60.04 The order satisfies the following ordering constraints: 213.53/60.04 213.53/60.04 [active(zeros())] = [0] 213.53/60.04 [0] 213.53/60.04 >= [0] 213.53/60.04 [0] 213.53/60.04 = [mark(cons(0(), zeros()))] 213.53/60.04 213.53/60.04 [active(tail(cons(X, XS)))] = [1 0] X + [1 1] XS + [0] 213.53/60.04 [0 1] [0 1] [1] 213.53/60.04 >= [1 1] XS + [0] 213.53/60.04 [0 1] [0] 213.53/60.04 = [mark(XS)] 213.53/60.04 213.53/60.04 [mark(zeros())] = [0] 213.53/60.04 [0] 213.53/60.04 >= [0] 213.53/60.04 [0] 213.53/60.04 = [active(zeros())] 213.53/60.04 213.53/60.04 [mark(cons(X1, X2))] = [1 1] X1 + [1 2] X2 + [0] 213.53/60.04 [0 1] [0 1] [0] 213.53/60.04 >= [1 1] X1 + [1 1] X2 + [0] 213.53/60.04 [0 1] [0 1] [0] 213.53/60.04 = [active(cons(mark(X1), X2))] 213.53/60.04 213.53/60.04 [mark(0())] = [0] 213.53/60.04 [0] 213.53/60.04 >= [0] 213.53/60.04 [0] 213.53/60.04 = [active(0())] 213.53/60.04 213.53/60.04 [mark(tail(X))] = [1 1] X + [1] 213.53/60.04 [0 1] [1] 213.53/60.04 > [1 1] X + [0] 213.53/60.04 [0 1] [1] 213.53/60.04 = [active(tail(mark(X)))] 213.53/60.04 213.53/60.04 [cons(X1, active(X2))] = [1 0] X1 + [1 1] X2 + [0] 213.53/60.04 [0 1] [0 1] [0] 213.53/60.04 >= [1 0] X1 + [1 1] X2 + [0] 213.53/60.04 [0 1] [0 1] [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(X1, mark(X2))] = [1 0] X1 + [1 2] X2 + [0] 213.53/60.04 [0 1] [0 1] [0] 213.53/60.04 >= [1 0] X1 + [1 1] X2 + [0] 213.53/60.04 [0 1] [0 1] [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(active(X1), X2)] = [1 0] X1 + [1 1] X2 + [0] 213.53/60.04 [0 1] [0 1] [0] 213.53/60.04 >= [1 0] X1 + [1 1] X2 + [0] 213.53/60.04 [0 1] [0 1] [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(mark(X1), X2)] = [1 1] X1 + [1 1] X2 + [0] 213.53/60.04 [0 1] [0 1] [0] 213.53/60.04 >= [1 0] X1 + [1 1] X2 + [0] 213.53/60.04 [0 1] [0 1] [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [tail(active(X))] = [1 0] X + [0] 213.53/60.04 [0 1] [1] 213.53/60.04 >= [1 0] X + [0] 213.53/60.04 [0 1] [1] 213.53/60.04 = [tail(X)] 213.53/60.04 213.53/60.04 [tail(mark(X))] = [1 1] X + [0] 213.53/60.04 [0 1] [1] 213.53/60.04 >= [1 0] X + [0] 213.53/60.04 [0 1] [1] 213.53/60.04 = [tail(X)] 213.53/60.04 213.53/60.04 213.53/60.04 We return to the main proof. 213.53/60.04 213.53/60.04 We are left with following problem, upon which TcT provides the 213.53/60.04 certificate YES(O(1),O(n^3)). 213.53/60.04 213.53/60.04 Strict Trs: 213.53/60.04 { active(zeros()) -> mark(cons(0(), zeros())) 213.53/60.04 , mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) } 213.53/60.04 Weak Trs: 213.53/60.04 { active(tail(cons(X, XS))) -> mark(XS) 213.53/60.04 , mark(zeros()) -> active(zeros()) 213.53/60.04 , mark(0()) -> active(0()) 213.53/60.04 , mark(tail(X)) -> active(tail(mark(X))) 213.53/60.04 , cons(X1, active(X2)) -> cons(X1, X2) 213.53/60.04 , cons(X1, mark(X2)) -> cons(X1, X2) 213.53/60.04 , cons(active(X1), X2) -> cons(X1, X2) 213.53/60.04 , cons(mark(X1), X2) -> cons(X1, X2) 213.53/60.04 , tail(active(X)) -> tail(X) 213.53/60.04 , tail(mark(X)) -> tail(X) } 213.53/60.04 Obligation: 213.53/60.04 derivational complexity 213.53/60.04 Answer: 213.53/60.04 YES(O(1),O(n^3)) 213.53/60.04 213.53/60.04 We use the processor 'matrix interpretation of dimension 3' to 213.53/60.04 orient following rules strictly. 213.53/60.04 213.53/60.04 Trs: { active(zeros()) -> mark(cons(0(), zeros())) } 213.53/60.04 213.53/60.04 The induced complexity on above rules (modulo remaining rules) is 213.53/60.04 YES(?,O(n^2)) . These rules are moved into the corresponding weak 213.53/60.04 component(s). 213.53/60.04 213.53/60.04 Sub-proof: 213.53/60.04 ---------- 213.53/60.04 TcT has computed the following triangular matrix interpretation. 213.53/60.04 Note that the diagonal of the component-wise maxima of 213.53/60.04 interpretation-entries contains no more than 2 non-zero entries. 213.53/60.04 213.53/60.04 [1 0 1] [0] 213.53/60.04 [active](x1) = [0 1 1] x1 + [0] 213.53/60.04 [0 0 0] [0] 213.53/60.04 213.53/60.04 [0] 213.53/60.04 [zeros] = [0] 213.53/60.04 [1] 213.53/60.04 213.53/60.04 [1 0 1] [0] 213.53/60.04 [mark](x1) = [0 1 1] x1 + [0] 213.53/60.04 [0 0 0] [0] 213.53/60.04 213.53/60.04 [1 1 2] [1 0 0] [0] 213.53/60.04 [cons](x1, x2) = [0 0 0] x1 + [0 1 1] x2 + [0] 213.53/60.04 [0 0 0] [0 0 0] [0] 213.53/60.04 213.53/60.04 [0] 213.53/60.04 [0] = [0] 213.53/60.04 [0] 213.53/60.04 213.53/60.04 [1 1 2] [0] 213.53/60.04 [tail](x1) = [0 1 1] x1 + [0] 213.53/60.04 [0 0 0] [0] 213.53/60.04 213.53/60.04 The order satisfies the following ordering constraints: 213.53/60.04 213.53/60.04 [active(zeros())] = [1] 213.53/60.04 [1] 213.53/60.04 [0] 213.53/60.04 > [0] 213.53/60.04 [1] 213.53/60.04 [0] 213.53/60.04 = [mark(cons(0(), zeros()))] 213.53/60.04 213.53/60.04 [active(tail(cons(X, XS)))] = [1 1 2] [1 1 1] [0] 213.53/60.04 [0 0 0] X + [0 1 1] XS + [0] 213.53/60.04 [0 0 0] [0 0 0] [0] 213.53/60.04 >= [1 0 1] [0] 213.53/60.04 [0 1 1] XS + [0] 213.53/60.04 [0 0 0] [0] 213.53/60.04 = [mark(XS)] 213.53/60.04 213.53/60.04 [mark(zeros())] = [1] 213.53/60.04 [1] 213.53/60.04 [0] 213.53/60.04 >= [1] 213.53/60.04 [1] 213.53/60.04 [0] 213.53/60.04 = [active(zeros())] 213.53/60.04 213.53/60.04 [mark(cons(X1, X2))] = [1 1 2] [1 0 0] [0] 213.53/60.04 [0 0 0] X1 + [0 1 1] X2 + [0] 213.53/60.04 [0 0 0] [0 0 0] [0] 213.53/60.04 >= [1 1 2] [1 0 0] [0] 213.53/60.04 [0 0 0] X1 + [0 1 1] X2 + [0] 213.53/60.04 [0 0 0] [0 0 0] [0] 213.53/60.04 = [active(cons(mark(X1), X2))] 213.53/60.04 213.53/60.04 [mark(0())] = [0] 213.53/60.04 [0] 213.53/60.04 [0] 213.53/60.04 >= [0] 213.53/60.04 [0] 213.53/60.04 [0] 213.53/60.04 = [active(0())] 213.53/60.04 213.53/60.04 [mark(tail(X))] = [1 1 2] [0] 213.53/60.04 [0 1 1] X + [0] 213.53/60.04 [0 0 0] [0] 213.53/60.04 >= [1 1 2] [0] 213.53/60.04 [0 1 1] X + [0] 213.53/60.04 [0 0 0] [0] 213.53/60.04 = [active(tail(mark(X)))] 213.53/60.04 213.53/60.04 [cons(X1, active(X2))] = [1 1 2] [1 0 1] [0] 213.53/60.04 [0 0 0] X1 + [0 1 1] X2 + [0] 213.53/60.04 [0 0 0] [0 0 0] [0] 213.53/60.04 >= [1 1 2] [1 0 0] [0] 213.53/60.04 [0 0 0] X1 + [0 1 1] X2 + [0] 213.53/60.04 [0 0 0] [0 0 0] [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(X1, mark(X2))] = [1 1 2] [1 0 1] [0] 213.53/60.04 [0 0 0] X1 + [0 1 1] X2 + [0] 213.53/60.04 [0 0 0] [0 0 0] [0] 213.53/60.04 >= [1 1 2] [1 0 0] [0] 213.53/60.04 [0 0 0] X1 + [0 1 1] X2 + [0] 213.53/60.04 [0 0 0] [0 0 0] [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(active(X1), X2)] = [1 1 2] [1 0 0] [0] 213.53/60.04 [0 0 0] X1 + [0 1 1] X2 + [0] 213.53/60.04 [0 0 0] [0 0 0] [0] 213.53/60.04 >= [1 1 2] [1 0 0] [0] 213.53/60.04 [0 0 0] X1 + [0 1 1] X2 + [0] 213.53/60.04 [0 0 0] [0 0 0] [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(mark(X1), X2)] = [1 1 2] [1 0 0] [0] 213.53/60.04 [0 0 0] X1 + [0 1 1] X2 + [0] 213.53/60.04 [0 0 0] [0 0 0] [0] 213.53/60.04 >= [1 1 2] [1 0 0] [0] 213.53/60.04 [0 0 0] X1 + [0 1 1] X2 + [0] 213.53/60.04 [0 0 0] [0 0 0] [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [tail(active(X))] = [1 1 2] [0] 213.53/60.04 [0 1 1] X + [0] 213.53/60.04 [0 0 0] [0] 213.53/60.04 >= [1 1 2] [0] 213.53/60.04 [0 1 1] X + [0] 213.53/60.04 [0 0 0] [0] 213.53/60.04 = [tail(X)] 213.53/60.04 213.53/60.04 [tail(mark(X))] = [1 1 2] [0] 213.53/60.04 [0 1 1] X + [0] 213.53/60.04 [0 0 0] [0] 213.53/60.04 >= [1 1 2] [0] 213.53/60.04 [0 1 1] X + [0] 213.53/60.04 [0 0 0] [0] 213.53/60.04 = [tail(X)] 213.53/60.04 213.53/60.04 213.53/60.04 We return to the main proof. 213.53/60.04 213.53/60.04 We are left with following problem, upon which TcT provides the 213.53/60.04 certificate YES(?,O(n^3)). 213.53/60.04 213.53/60.04 Strict Trs: { mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) } 213.53/60.04 Weak Trs: 213.53/60.04 { active(zeros()) -> mark(cons(0(), zeros())) 213.53/60.04 , active(tail(cons(X, XS))) -> mark(XS) 213.53/60.04 , mark(zeros()) -> active(zeros()) 213.53/60.04 , mark(0()) -> active(0()) 213.53/60.04 , mark(tail(X)) -> active(tail(mark(X))) 213.53/60.04 , cons(X1, active(X2)) -> cons(X1, X2) 213.53/60.04 , cons(X1, mark(X2)) -> cons(X1, X2) 213.53/60.04 , cons(active(X1), X2) -> cons(X1, X2) 213.53/60.04 , cons(mark(X1), X2) -> cons(X1, X2) 213.53/60.04 , tail(active(X)) -> tail(X) 213.53/60.04 , tail(mark(X)) -> tail(X) } 213.53/60.04 Obligation: 213.53/60.04 derivational complexity 213.53/60.04 Answer: 213.53/60.04 YES(?,O(n^3)) 213.53/60.04 213.53/60.04 TcT has computed the following triangular matrix interpretation. 213.53/60.04 213.53/60.04 [1 0 0 3] [0] 213.53/60.04 [active](x1) = [0 1 0 7] x1 + [0] 213.53/60.04 [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0] 213.53/60.04 213.53/60.04 [0] 213.53/60.04 [zeros] = [1] 213.53/60.04 [0] 213.53/60.04 [1] 213.53/60.04 213.53/60.04 [1 1 0 2] [0] 213.53/60.04 [mark](x1) = [0 1 0 2] x1 + [5] 213.53/60.04 [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0] 213.53/60.04 213.53/60.04 [1 0 3 4] [1 1 0 0] [0] 213.53/60.04 [cons](x1, x2) = [0 1 1 4] x1 + [0 1 2 0] x2 + [1] 213.53/60.04 [0 0 0 0] [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0 0 0 0] [0] 213.53/60.04 213.53/60.04 [0] 213.53/60.04 [0] = [0] 213.53/60.04 [0] 213.53/60.04 [0] 213.53/60.04 213.53/60.04 [1 0 4 4] [0] 213.53/60.04 [tail](x1) = [0 1 2 6] x1 + [4] 213.53/60.04 [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0] 213.53/60.04 213.53/60.04 The order satisfies the following ordering constraints: 213.53/60.04 213.53/60.04 [active(zeros())] = [3] 213.53/60.04 [8] 213.53/60.04 [2] 213.53/60.04 [0] 213.53/60.04 >= [3] 213.53/60.04 [7] 213.53/60.04 [2] 213.53/60.04 [0] 213.53/60.04 = [mark(cons(0(), zeros()))] 213.53/60.04 213.53/60.04 [active(tail(cons(X, XS)))] = [1 0 3 4] [1 1 4 8] [0] 213.53/60.04 [0 1 1 4] X + [0 1 4 4] XS + [5] 213.53/60.04 [0 0 0 0] [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0 0 0 0] [0] 213.53/60.04 >= [1 1 0 2] [0] 213.53/60.04 [0 1 0 2] XS + [5] 213.53/60.04 [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0] 213.53/60.04 = [mark(XS)] 213.53/60.04 213.53/60.04 [mark(zeros())] = [3] 213.53/60.04 [8] 213.53/60.04 [2] 213.53/60.04 [0] 213.53/60.04 >= [3] 213.53/60.04 [8] 213.53/60.04 [2] 213.53/60.04 [0] 213.53/60.04 = [active(zeros())] 213.53/60.04 213.53/60.04 [mark(cons(X1, X2))] = [1 1 4 8] [1 2 2 0] [1] 213.53/60.04 [0 1 1 4] X1 + [0 1 2 0] X2 + [6] 213.53/60.04 [0 0 0 0] [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0 0 0 0] [0] 213.53/60.04 > [1 1 3 8] [1 1 0 0] [0] 213.53/60.04 [0 1 1 4] X1 + [0 1 2 0] X2 + [6] 213.53/60.04 [0 0 0 0] [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0 0 0 0] [0] 213.53/60.04 = [active(cons(mark(X1), X2))] 213.53/60.04 213.53/60.04 [mark(0())] = [0] 213.53/60.04 [5] 213.53/60.04 [0] 213.53/60.04 [0] 213.53/60.04 >= [0] 213.53/60.04 [0] 213.53/60.04 [0] 213.53/60.04 [0] 213.53/60.04 = [active(0())] 213.53/60.04 213.53/60.04 [mark(tail(X))] = [1 1 6 10] [4] 213.53/60.04 [0 1 2 6] X + [9] 213.53/60.04 [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0] 213.53/60.04 > [1 1 4 10] [0] 213.53/60.04 [0 1 2 6] X + [9] 213.53/60.04 [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0] 213.53/60.04 = [active(tail(mark(X)))] 213.53/60.04 213.53/60.04 [cons(X1, active(X2))] = [1 0 3 4] [1 1 0 10] [0] 213.53/60.04 [0 1 1 4] X1 + [0 1 2 11] X2 + [1] 213.53/60.04 [0 0 0 0] [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0 0 0 0] [0] 213.53/60.04 >= [1 0 3 4] [1 1 0 0] [0] 213.53/60.04 [0 1 1 4] X1 + [0 1 2 0] X2 + [1] 213.53/60.04 [0 0 0 0] [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0 0 0 0] [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(X1, mark(X2))] = [1 0 3 4] [1 2 0 4] [5] 213.53/60.04 [0 1 1 4] X1 + [0 1 2 6] X2 + [6] 213.53/60.04 [0 0 0 0] [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0 0 0 0] [0] 213.53/60.04 > [1 0 3 4] [1 1 0 0] [0] 213.53/60.04 [0 1 1 4] X1 + [0 1 2 0] X2 + [1] 213.53/60.04 [0 0 0 0] [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0 0 0 0] [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(active(X1), X2)] = [1 0 3 9] [1 1 0 0] [0] 213.53/60.04 [0 1 1 9] X1 + [0 1 2 0] X2 + [1] 213.53/60.04 [0 0 0 0] [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0 0 0 0] [0] 213.53/60.04 >= [1 0 3 4] [1 1 0 0] [0] 213.53/60.04 [0 1 1 4] X1 + [0 1 2 0] X2 + [1] 213.53/60.04 [0 0 0 0] [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0 0 0 0] [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [cons(mark(X1), X2)] = [1 1 3 8] [1 1 0 0] [0] 213.53/60.04 [0 1 1 4] X1 + [0 1 2 0] X2 + [6] 213.53/60.04 [0 0 0 0] [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0 0 0 0] [0] 213.53/60.04 >= [1 0 3 4] [1 1 0 0] [0] 213.53/60.04 [0 1 1 4] X1 + [0 1 2 0] X2 + [1] 213.53/60.04 [0 0 0 0] [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0 0 0 0] [0] 213.53/60.04 = [cons(X1, X2)] 213.53/60.04 213.53/60.04 [tail(active(X))] = [1 0 4 11] [0] 213.53/60.04 [0 1 2 11] X + [4] 213.53/60.04 [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0] 213.53/60.04 >= [1 0 4 4] [0] 213.53/60.04 [0 1 2 6] X + [4] 213.53/60.04 [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0] 213.53/60.04 = [tail(X)] 213.53/60.04 213.53/60.04 [tail(mark(X))] = [1 1 4 10] [0] 213.53/60.04 [0 1 2 6] X + [9] 213.53/60.04 [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0] 213.53/60.04 >= [1 0 4 4] [0] 213.53/60.04 [0 1 2 6] X + [4] 213.53/60.04 [0 0 1 2] [0] 213.53/60.04 [0 0 0 0] [0] 213.53/60.04 = [tail(X)] 213.53/60.04 213.53/60.04 213.53/60.04 Hurray, we answered YES(O(1),O(n^3)) 213.53/60.08 EOF