YES(O(1),O(n^2)) 165.52/60.07 YES(O(1),O(n^2)) 165.52/60.07 165.52/60.07 We are left with following problem, upon which TcT provides the 165.52/60.07 certificate YES(O(1),O(n^2)). 165.52/60.07 165.52/60.07 Strict Trs: 165.52/60.07 { __(X, nil()) -> X 165.52/60.07 , __(__(X, Y), Z) -> __(X, __(Y, Z)) 165.52/60.07 , __(nil(), X) -> X 165.52/60.07 , and(tt(), X) -> activate(X) 165.52/60.07 , activate(X) -> X 165.52/60.07 , isNePal(__(I, __(P, I))) -> tt() } 165.52/60.07 Obligation: 165.52/60.07 derivational complexity 165.52/60.07 Answer: 165.52/60.07 YES(O(1),O(n^2)) 165.52/60.07 165.52/60.07 We use the processor 'matrix interpretation of dimension 1' to 165.52/60.07 orient following rules strictly. 165.52/60.07 165.52/60.07 Trs: 165.52/60.07 { __(X, nil()) -> X 165.52/60.07 , __(nil(), X) -> X 165.52/60.07 , and(tt(), X) -> activate(X) 165.52/60.07 , activate(X) -> X 165.52/60.07 , isNePal(__(I, __(P, I))) -> tt() } 165.52/60.07 165.52/60.07 The induced complexity on above rules (modulo remaining rules) is 165.52/60.07 YES(?,O(n^1)) . These rules are moved into the corresponding weak 165.52/60.07 component(s). 165.52/60.07 165.52/60.07 Sub-proof: 165.52/60.07 ---------- 165.52/60.07 TcT has computed the following triangular matrix interpretation. 165.52/60.07 165.52/60.07 [__](x1, x2) = [1] x1 + [1] x2 + [2] 165.52/60.07 165.52/60.07 [nil] = [2] 165.52/60.07 165.52/60.07 [and](x1, x2) = [1] x1 + [1] x2 + [2] 165.52/60.07 165.52/60.07 [tt] = [1] 165.52/60.07 165.52/60.07 [activate](x1) = [1] x1 + [1] 165.52/60.07 165.52/60.07 [isNePal](x1) = [1] x1 + [2] 165.52/60.07 165.52/60.07 The order satisfies the following ordering constraints: 165.52/60.07 165.52/60.07 [__(X, nil())] = [1] X + [4] 165.52/60.07 > [1] X + [0] 165.52/60.07 = [X] 165.52/60.07 165.52/60.07 [__(__(X, Y), Z)] = [1] X + [1] Y + [1] Z + [4] 165.52/60.07 >= [1] X + [1] Y + [1] Z + [4] 165.52/60.07 = [__(X, __(Y, Z))] 165.52/60.07 165.52/60.07 [__(nil(), X)] = [1] X + [4] 165.52/60.07 > [1] X + [0] 165.52/60.07 = [X] 165.52/60.07 165.52/60.07 [and(tt(), X)] = [1] X + [3] 165.52/60.07 > [1] X + [1] 165.52/60.07 = [activate(X)] 165.52/60.07 165.52/60.07 [activate(X)] = [1] X + [1] 165.52/60.07 > [1] X + [0] 165.52/60.07 = [X] 165.52/60.07 165.52/60.07 [isNePal(__(I, __(P, I)))] = [2] I + [1] P + [6] 165.52/60.07 > [1] 165.52/60.07 = [tt()] 165.52/60.07 165.52/60.07 165.52/60.07 We return to the main proof. 165.52/60.07 165.52/60.07 We are left with following problem, upon which TcT provides the 165.52/60.07 certificate YES(O(1),O(n^2)). 165.52/60.07 165.52/60.07 Strict Trs: { __(__(X, Y), Z) -> __(X, __(Y, Z)) } 165.52/60.07 Weak Trs: 165.52/60.07 { __(X, nil()) -> X 165.52/60.07 , __(nil(), X) -> X 165.52/60.07 , and(tt(), X) -> activate(X) 165.52/60.07 , activate(X) -> X 165.52/60.07 , isNePal(__(I, __(P, I))) -> tt() } 165.52/60.07 Obligation: 165.52/60.07 derivational complexity 165.52/60.07 Answer: 165.52/60.07 YES(O(1),O(n^2)) 165.52/60.07 165.52/60.07 We use the processor 'matrix interpretation of dimension 2' to 165.52/60.07 orient following rules strictly. 165.52/60.07 165.52/60.07 Trs: { __(__(X, Y), Z) -> __(X, __(Y, Z)) } 165.52/60.07 165.52/60.07 The induced complexity on above rules (modulo remaining rules) is 165.52/60.07 YES(?,O(n^2)) . These rules are moved into the corresponding weak 165.52/60.07 component(s). 165.52/60.07 165.52/60.07 Sub-proof: 165.52/60.07 ---------- 165.52/60.07 TcT has computed the following triangular matrix interpretation. 165.52/60.07 165.52/60.07 [__](x1, x2) = [1 2] x1 + [1 0] x2 + [0] 165.52/60.07 [0 1] [0 1] [2] 165.52/60.07 165.52/60.07 [nil] = [1] 165.52/60.07 [1] 165.52/60.07 165.52/60.07 [and](x1, x2) = [1 1] x1 + [1 1] x2 + [1] 165.52/60.07 [0 1] [0 1] [1] 165.52/60.07 165.52/60.07 [tt] = [1] 165.52/60.07 [1] 165.52/60.07 165.52/60.07 [activate](x1) = [1 1] x1 + [1] 165.52/60.07 [0 1] [1] 165.52/60.07 165.52/60.07 [isNePal](x1) = [1 0] x1 + [2] 165.52/60.07 [0 1] [2] 165.52/60.07 165.52/60.07 The order satisfies the following ordering constraints: 165.52/60.07 165.52/60.07 [__(X, nil())] = [1 2] X + [1] 165.52/60.07 [0 1] [3] 165.52/60.07 > [1 0] X + [0] 165.52/60.07 [0 1] [0] 165.52/60.07 = [X] 165.52/60.07 165.52/60.07 [__(__(X, Y), Z)] = [1 4] X + [1 2] Y + [1 0] Z + [4] 165.52/60.07 [0 1] [0 1] [0 1] [4] 165.52/60.07 > [1 2] X + [1 2] Y + [1 0] Z + [0] 165.52/60.07 [0 1] [0 1] [0 1] [4] 165.52/60.07 = [__(X, __(Y, Z))] 165.52/60.07 165.52/60.07 [__(nil(), X)] = [1 0] X + [3] 165.52/60.07 [0 1] [3] 165.52/60.07 > [1 0] X + [0] 165.52/60.07 [0 1] [0] 165.52/60.07 = [X] 165.52/60.07 165.52/60.07 [and(tt(), X)] = [1 1] X + [3] 165.52/60.07 [0 1] [2] 165.52/60.07 > [1 1] X + [1] 165.52/60.07 [0 1] [1] 165.52/60.07 = [activate(X)] 165.52/60.07 165.52/60.07 [activate(X)] = [1 1] X + [1] 165.52/60.07 [0 1] [1] 165.52/60.07 > [1 0] X + [0] 165.52/60.07 [0 1] [0] 165.52/60.07 = [X] 165.52/60.07 165.52/60.07 [isNePal(__(I, __(P, I)))] = [2 2] I + [1 2] P + [2] 165.52/60.07 [0 2] [0 1] [6] 165.52/60.07 > [1] 165.52/60.07 [1] 165.52/60.07 = [tt()] 165.52/60.07 165.52/60.07 165.52/60.07 We return to the main proof. 165.52/60.07 165.52/60.07 We are left with following problem, upon which TcT provides the 165.52/60.07 certificate YES(O(1),O(1)). 165.52/60.07 165.52/60.07 Weak Trs: 165.52/60.07 { __(X, nil()) -> X 165.52/60.07 , __(__(X, Y), Z) -> __(X, __(Y, Z)) 165.52/60.07 , __(nil(), X) -> X 165.52/60.07 , and(tt(), X) -> activate(X) 165.52/60.07 , activate(X) -> X 165.52/60.07 , isNePal(__(I, __(P, I))) -> tt() } 165.52/60.07 Obligation: 165.52/60.07 derivational complexity 165.52/60.07 Answer: 165.52/60.07 YES(O(1),O(1)) 165.52/60.07 165.52/60.07 Empty rules are trivially bounded 165.52/60.07 165.52/60.07 Hurray, we answered YES(O(1),O(n^2)) 165.73/60.10 EOF