YES(O(1),O(n^6)) 281.90/83.72 YES(O(1),O(n^6)) 281.90/83.72 281.90/83.72 We are left with following problem, upon which TcT provides the 281.90/83.72 certificate YES(O(1),O(n^6)). 281.90/83.72 281.90/83.72 Strict Trs: 281.90/83.72 { app(l, nil()) -> l 281.90/83.72 , app(nil(), k) -> k 281.90/83.72 , app(cons(x, l), k) -> cons(x, app(l, k)) 281.90/83.72 , sum(app(l, cons(x, cons(y, k)))) -> 281.90/83.72 sum(app(l, sum(cons(x, cons(y, k))))) 281.90/83.72 , sum(cons(x, nil())) -> cons(x, nil()) 281.90/83.72 , sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)) 281.90/83.72 , plus(0(), y) -> y 281.90/83.72 , plus(s(x), y) -> s(plus(x, y)) } 281.90/83.72 Obligation: 281.90/83.72 derivational complexity 281.90/83.72 Answer: 281.90/83.72 YES(O(1),O(n^6)) 281.90/83.72 281.90/83.72 We use the processor 'matrix interpretation of dimension 2' to 281.90/83.72 orient following rules strictly. 281.90/83.72 281.90/83.72 Trs: 281.90/83.72 { app(l, nil()) -> l 281.90/83.72 , app(nil(), k) -> k 281.90/83.72 , app(cons(x, l), k) -> cons(x, app(l, k)) 281.90/83.72 , sum(app(l, cons(x, cons(y, k)))) -> 281.90/83.72 sum(app(l, sum(cons(x, cons(y, k))))) 281.90/83.72 , sum(cons(x, nil())) -> cons(x, nil()) 281.90/83.72 , sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)) 281.90/83.72 , plus(0(), y) -> y } 281.90/83.72 281.90/83.72 The induced complexity on above rules (modulo remaining rules) is 281.90/83.72 YES(?,O(n^2)) . These rules are removed from the problem. Note that 281.90/83.72 none of the weakly oriented rules is size-increasing. The overall 281.90/83.72 complexity is obtained by composition . 281.90/83.72 281.90/83.72 Sub-proof: 281.90/83.72 ---------- 281.90/83.72 TcT has computed the following triangular matrix interpretation. 281.90/83.72 281.90/83.72 [app](x1, x2) = [1 1] x1 + [1 2] x2 + [0] 281.90/83.72 [0 1] [0 1] [0] 281.90/83.72 281.90/83.72 [nil] = [2] 281.90/83.72 [0] 281.90/83.72 281.90/83.72 [cons](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 281.90/83.72 [0 0] [0 1] [1] 281.90/83.72 281.90/83.72 [sum](x1) = [1 0] x1 + [1] 281.90/83.72 [0 0] [1] 281.90/83.72 281.90/83.72 [plus](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 281.90/83.72 [0 0] [0 1] [0] 281.90/83.72 281.90/83.72 [0] = [2] 281.90/83.72 [2] 281.90/83.72 281.90/83.72 [s](x1) = [1 0] x1 + [0] 281.90/83.72 [0 1] [0] 281.90/83.72 281.90/83.72 The order satisfies the following ordering constraints: 281.90/83.72 281.90/83.72 [app(l, nil())] = [1 1] l + [2] 281.90/83.72 [0 1] [0] 281.90/83.72 > [1 0] l + [0] 281.90/83.72 [0 1] [0] 281.90/83.72 = [l] 281.90/83.72 281.90/83.72 [app(nil(), k)] = [1 2] k + [2] 281.90/83.72 [0 1] [0] 281.90/83.72 > [1 0] k + [0] 281.90/83.72 [0 1] [0] 281.90/83.72 = [k] 281.90/83.72 281.90/83.72 [app(cons(x, l), k)] = [1 2] k + [1 1] l + [1 0] x + [2] 281.90/83.72 [0 1] [0 1] [0 0] [1] 281.90/83.72 > [1 2] k + [1 1] l + [1 0] x + [1] 281.90/83.72 [0 1] [0 1] [0 0] [1] 281.90/83.72 = [cons(x, app(l, k))] 281.90/83.72 281.90/83.72 [sum(app(l, cons(x, cons(y, k))))] = [1 2] k + [1 1] l + [1 0] x + [1 0] y + [7] 281.90/83.72 [0 0] [0 0] [0 0] [0 0] [1] 281.90/83.72 > [1 0] k + [1 1] l + [1 0] x + [1 0] y + [6] 281.90/83.72 [0 0] [0 0] [0 0] [0 0] [1] 281.90/83.73 = [sum(app(l, sum(cons(x, cons(y, k)))))] 281.90/83.73 281.90/83.73 [sum(cons(x, nil()))] = [1 0] x + [4] 281.90/83.73 [0 0] [1] 281.90/83.73 > [1 0] x + [3] 281.90/83.73 [0 0] [1] 281.90/83.73 = [cons(x, nil())] 281.90/83.73 281.90/83.73 [sum(cons(x, cons(y, l)))] = [1 0] l + [1 0] x + [1 0] y + [3] 281.90/83.73 [0 0] [0 0] [0 0] [1] 281.90/83.73 > [1 0] l + [1 0] x + [1 0] y + [2] 281.90/83.73 [0 0] [0 0] [0 0] [1] 281.90/83.73 = [sum(cons(plus(x, y), l))] 281.90/83.73 281.90/83.73 [plus(0(), y)] = [1 0] y + [2] 281.90/83.73 [0 1] [0] 281.90/83.73 > [1 0] y + [0] 281.90/83.73 [0 1] [0] 281.90/83.73 = [y] 281.90/83.73 281.90/83.73 [plus(s(x), y)] = [1 0] x + [1 0] y + [0] 281.90/83.73 [0 0] [0 1] [0] 281.90/83.73 >= [1 0] x + [1 0] y + [0] 281.90/83.73 [0 0] [0 1] [0] 281.90/83.73 = [s(plus(x, y))] 281.90/83.73 281.90/83.73 281.90/83.73 We return to the main proof. 281.90/83.73 281.90/83.73 We are left with following problem, upon which TcT provides the 281.90/83.73 certificate YES(?,O(n^2)). 281.90/83.73 281.90/83.73 Strict Trs: { plus(s(x), y) -> s(plus(x, y)) } 281.90/83.73 Obligation: 281.90/83.73 derivational complexity 281.90/83.73 Answer: 281.90/83.73 YES(?,O(n^2)) 281.90/83.73 281.90/83.73 TcT has computed the following triangular matrix interpretation. 281.90/83.73 281.90/83.73 [plus](x1, x2) = [1 4] x1 + [1 7] x2 + [0] 281.90/83.73 [0 1] [0 0] [0] 281.90/83.73 281.90/83.73 [s](x1) = [1 4] x1 + [0] 281.90/83.73 [0 1] [2] 281.90/83.73 281.90/83.73 The order satisfies the following ordering constraints: 281.90/83.73 281.90/83.73 [plus(s(x), y)] = [1 8] x + [1 7] y + [8] 281.90/83.73 [0 1] [0 0] [2] 281.90/83.73 > [1 8] x + [1 7] y + [0] 281.90/83.73 [0 1] [0 0] [2] 281.90/83.73 = [s(plus(x, y))] 281.90/83.73 281.90/83.73 281.90/83.73 Hurray, we answered YES(O(1),O(n^6)) 282.09/83.85 EOF