YES(O(1),O(n^2)) 164.94/60.06 YES(O(1),O(n^2)) 164.94/60.06 164.94/60.06 We are left with following problem, upon which TcT provides the 164.94/60.06 certificate YES(O(1),O(n^2)). 164.94/60.06 164.94/60.06 Strict Trs: 164.94/60.06 { __(X, nil()) -> X 164.94/60.06 , __(__(X, Y), Z) -> __(X, __(Y, Z)) 164.94/60.06 , __(nil(), X) -> X 164.94/60.06 , U11(tt()) -> U12(tt()) 164.94/60.06 , U12(tt()) -> tt() 164.94/60.06 , isNePal(__(I, __(P, I))) -> U11(tt()) 164.94/60.06 , activate(X) -> X } 164.94/60.06 Obligation: 164.94/60.06 derivational complexity 164.94/60.06 Answer: 164.94/60.06 YES(O(1),O(n^2)) 164.94/60.06 164.94/60.06 We use the processor 'matrix interpretation of dimension 1' to 164.94/60.06 orient following rules strictly. 164.94/60.06 164.94/60.06 Trs: 164.94/60.06 { __(X, nil()) -> X 164.94/60.06 , __(nil(), X) -> X 164.94/60.06 , isNePal(__(I, __(P, I))) -> U11(tt()) 164.94/60.06 , activate(X) -> X } 164.94/60.06 164.94/60.06 The induced complexity on above rules (modulo remaining rules) is 164.94/60.06 YES(?,O(n^1)) . These rules are moved into the corresponding weak 164.94/60.06 component(s). 164.94/60.06 164.94/60.06 Sub-proof: 164.94/60.06 ---------- 164.94/60.06 TcT has computed the following triangular matrix interpretation. 164.94/60.06 164.94/60.06 [__](x1, x2) = [1] x1 + [1] x2 + [2] 164.94/60.06 164.94/60.06 [nil] = [1] 164.94/60.06 164.94/60.06 [U11](x1) = [1] x1 + [0] 164.94/60.06 164.94/60.06 [tt] = [0] 164.94/60.06 164.94/60.06 [U12](x1) = [1] x1 + [0] 164.94/60.06 164.94/60.06 [isNePal](x1) = [1] x1 + [2] 164.94/60.06 164.94/60.06 [activate](x1) = [1] x1 + [2] 164.94/60.06 164.94/60.06 The order satisfies the following ordering constraints: 164.94/60.06 164.94/60.06 [__(X, nil())] = [1] X + [3] 164.94/60.06 > [1] X + [0] 164.94/60.06 = [X] 164.94/60.06 164.94/60.06 [__(__(X, Y), Z)] = [1] X + [1] Y + [1] Z + [4] 164.94/60.06 >= [1] X + [1] Y + [1] Z + [4] 164.94/60.06 = [__(X, __(Y, Z))] 164.94/60.06 164.94/60.06 [__(nil(), X)] = [1] X + [3] 164.94/60.06 > [1] X + [0] 164.94/60.06 = [X] 164.94/60.06 164.94/60.06 [U11(tt())] = [0] 164.94/60.06 >= [0] 164.94/60.06 = [U12(tt())] 164.94/60.06 164.94/60.06 [U12(tt())] = [0] 164.94/60.06 >= [0] 164.94/60.06 = [tt()] 164.94/60.06 164.94/60.06 [isNePal(__(I, __(P, I)))] = [2] I + [1] P + [6] 164.94/60.06 > [0] 164.94/60.06 = [U11(tt())] 164.94/60.06 164.94/60.06 [activate(X)] = [1] X + [2] 164.94/60.06 > [1] X + [0] 164.94/60.06 = [X] 164.94/60.06 164.94/60.06 164.94/60.06 We return to the main proof. 164.94/60.06 164.94/60.06 We are left with following problem, upon which TcT provides the 164.94/60.06 certificate YES(O(1),O(n^2)). 164.94/60.06 164.94/60.06 Strict Trs: 164.94/60.06 { __(__(X, Y), Z) -> __(X, __(Y, Z)) 164.94/60.06 , U11(tt()) -> U12(tt()) 164.94/60.06 , U12(tt()) -> tt() } 164.94/60.06 Weak Trs: 164.94/60.06 { __(X, nil()) -> X 164.94/60.06 , __(nil(), X) -> X 164.94/60.06 , isNePal(__(I, __(P, I))) -> U11(tt()) 164.94/60.06 , activate(X) -> X } 164.94/60.06 Obligation: 164.94/60.06 derivational complexity 164.94/60.06 Answer: 164.94/60.06 YES(O(1),O(n^2)) 164.94/60.06 164.94/60.06 The weightgap principle applies (using the following nonconstant 164.94/60.06 growth matrix-interpretation) 164.94/60.06 164.94/60.06 TcT has computed the following triangular matrix interpretation. 164.94/60.06 Note that the diagonal of the component-wise maxima of 164.94/60.06 interpretation-entries contains no more than 1 non-zero entries. 164.94/60.06 164.94/60.06 [__](x1, x2) = [1] x1 + [1] x2 + [2] 164.94/60.06 164.94/60.06 [nil] = [1] 164.94/60.06 164.94/60.06 [U11](x1) = [1] x1 + [0] 164.94/60.06 164.94/60.06 [tt] = [0] 164.94/60.06 164.94/60.06 [U12](x1) = [1] x1 + [1] 164.94/60.06 164.94/60.06 [isNePal](x1) = [1] x1 + [2] 164.94/60.06 164.94/60.06 [activate](x1) = [1] x1 + [1] 164.94/60.06 164.94/60.06 The order satisfies the following ordering constraints: 164.94/60.06 164.94/60.06 [__(X, nil())] = [1] X + [3] 164.94/60.06 > [1] X + [0] 164.94/60.06 = [X] 164.94/60.06 164.94/60.06 [__(__(X, Y), Z)] = [1] X + [1] Y + [1] Z + [4] 164.94/60.06 >= [1] X + [1] Y + [1] Z + [4] 164.94/60.06 = [__(X, __(Y, Z))] 164.94/60.06 164.94/60.06 [__(nil(), X)] = [1] X + [3] 164.94/60.06 > [1] X + [0] 164.94/60.06 = [X] 164.94/60.06 164.94/60.06 [U11(tt())] = [0] 164.94/60.06 ? [1] 164.94/60.06 = [U12(tt())] 164.94/60.06 164.94/60.06 [U12(tt())] = [1] 164.94/60.06 > [0] 164.94/60.06 = [tt()] 164.94/60.06 164.94/60.06 [isNePal(__(I, __(P, I)))] = [2] I + [1] P + [6] 164.94/60.06 > [0] 164.94/60.06 = [U11(tt())] 164.94/60.06 164.94/60.06 [activate(X)] = [1] X + [1] 164.94/60.06 > [1] X + [0] 164.94/60.06 = [X] 164.94/60.06 164.94/60.06 164.94/60.06 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 164.94/60.06 164.94/60.06 We are left with following problem, upon which TcT provides the 164.94/60.06 certificate YES(O(1),O(n^2)). 164.94/60.06 164.94/60.06 Strict Trs: 164.94/60.06 { __(__(X, Y), Z) -> __(X, __(Y, Z)) 164.94/60.06 , U11(tt()) -> U12(tt()) } 164.94/60.06 Weak Trs: 164.94/60.06 { __(X, nil()) -> X 164.94/60.06 , __(nil(), X) -> X 164.94/60.06 , U12(tt()) -> tt() 164.94/60.06 , isNePal(__(I, __(P, I))) -> U11(tt()) 164.94/60.06 , activate(X) -> X } 164.94/60.06 Obligation: 164.94/60.06 derivational complexity 164.94/60.06 Answer: 164.94/60.06 YES(O(1),O(n^2)) 164.94/60.06 164.94/60.06 We use the processor 'matrix interpretation of dimension 1' to 164.94/60.06 orient following rules strictly. 164.94/60.06 164.94/60.06 Trs: { U11(tt()) -> U12(tt()) } 164.94/60.06 164.94/60.06 The induced complexity on above rules (modulo remaining rules) is 164.94/60.06 YES(?,O(n^1)) . These rules are moved into the corresponding weak 164.94/60.06 component(s). 164.94/60.06 164.94/60.06 Sub-proof: 164.94/60.06 ---------- 164.94/60.06 TcT has computed the following triangular matrix interpretation. 164.94/60.06 164.94/60.06 [__](x1, x2) = [1] x1 + [1] x2 + [2] 164.94/60.06 164.94/60.06 [nil] = [1] 164.94/60.06 164.94/60.06 [U11](x1) = [1] x1 + [1] 164.94/60.06 164.94/60.06 [tt] = [0] 164.94/60.06 164.94/60.06 [U12](x1) = [1] x1 + [0] 164.94/60.06 164.94/60.06 [isNePal](x1) = [1] x1 + [2] 164.94/60.06 164.94/60.06 [activate](x1) = [1] x1 + [1] 164.94/60.06 164.94/60.06 The order satisfies the following ordering constraints: 164.94/60.06 164.94/60.06 [__(X, nil())] = [1] X + [3] 164.94/60.06 > [1] X + [0] 164.94/60.06 = [X] 164.94/60.06 164.94/60.06 [__(__(X, Y), Z)] = [1] X + [1] Y + [1] Z + [4] 164.94/60.06 >= [1] X + [1] Y + [1] Z + [4] 164.94/60.06 = [__(X, __(Y, Z))] 164.94/60.06 164.94/60.06 [__(nil(), X)] = [1] X + [3] 164.94/60.06 > [1] X + [0] 164.94/60.06 = [X] 164.94/60.06 164.94/60.06 [U11(tt())] = [1] 164.94/60.06 > [0] 164.94/60.06 = [U12(tt())] 164.94/60.06 164.94/60.06 [U12(tt())] = [0] 164.94/60.06 >= [0] 164.94/60.06 = [tt()] 164.94/60.06 164.94/60.06 [isNePal(__(I, __(P, I)))] = [2] I + [1] P + [6] 164.94/60.06 > [1] 164.94/60.06 = [U11(tt())] 164.94/60.06 164.94/60.06 [activate(X)] = [1] X + [1] 164.94/60.06 > [1] X + [0] 164.94/60.06 = [X] 164.94/60.06 164.94/60.06 164.94/60.06 We return to the main proof. 164.94/60.06 164.94/60.06 We are left with following problem, upon which TcT provides the 164.94/60.06 certificate YES(O(1),O(n^2)). 164.94/60.06 164.94/60.06 Strict Trs: { __(__(X, Y), Z) -> __(X, __(Y, Z)) } 164.94/60.06 Weak Trs: 164.94/60.06 { __(X, nil()) -> X 164.94/60.06 , __(nil(), X) -> X 164.94/60.06 , U11(tt()) -> U12(tt()) 164.94/60.06 , U12(tt()) -> tt() 164.94/60.06 , isNePal(__(I, __(P, I))) -> U11(tt()) 164.94/60.06 , activate(X) -> X } 164.94/60.06 Obligation: 164.94/60.06 derivational complexity 164.94/60.06 Answer: 164.94/60.06 YES(O(1),O(n^2)) 164.94/60.06 164.94/60.06 We use the processor 'matrix interpretation of dimension 2' to 164.94/60.06 orient following rules strictly. 164.94/60.06 164.94/60.06 Trs: { __(__(X, Y), Z) -> __(X, __(Y, Z)) } 164.94/60.06 164.94/60.06 The induced complexity on above rules (modulo remaining rules) is 164.94/60.06 YES(?,O(n^2)) . These rules are moved into the corresponding weak 164.94/60.06 component(s). 164.94/60.06 164.94/60.06 Sub-proof: 164.94/60.06 ---------- 164.94/60.06 TcT has computed the following triangular matrix interpretation. 164.94/60.06 164.94/60.06 [__](x1, x2) = [1 2] x1 + [1 0] x2 + [0] 164.94/60.06 [0 1] [0 1] [2] 164.94/60.06 164.94/60.06 [nil] = [1] 164.94/60.06 [1] 164.94/60.06 164.94/60.06 [U11](x1) = [1 1] x1 + [0] 164.94/60.06 [0 1] [0] 164.94/60.06 164.94/60.06 [tt] = [0] 164.94/60.06 [0] 164.94/60.06 164.94/60.06 [U12](x1) = [1 2] x1 + [0] 164.94/60.06 [0 1] [0] 164.94/60.06 164.94/60.06 [isNePal](x1) = [1 0] x1 + [2] 164.94/60.06 [0 0] [2] 164.94/60.06 164.94/60.06 [activate](x1) = [1 1] x1 + [1] 164.94/60.06 [0 1] [1] 164.94/60.06 164.94/60.06 The order satisfies the following ordering constraints: 164.94/60.06 164.94/60.06 [__(X, nil())] = [1 2] X + [1] 164.94/60.06 [0 1] [3] 164.94/60.06 > [1 0] X + [0] 164.94/60.06 [0 1] [0] 164.94/60.06 = [X] 164.94/60.06 164.94/60.06 [__(__(X, Y), Z)] = [1 4] X + [1 2] Y + [1 0] Z + [4] 164.94/60.06 [0 1] [0 1] [0 1] [4] 164.94/60.06 > [1 2] X + [1 2] Y + [1 0] Z + [0] 164.94/60.06 [0 1] [0 1] [0 1] [4] 164.94/60.06 = [__(X, __(Y, Z))] 164.94/60.06 164.94/60.06 [__(nil(), X)] = [1 0] X + [3] 164.94/60.06 [0 1] [3] 164.94/60.06 > [1 0] X + [0] 164.94/60.06 [0 1] [0] 164.94/60.06 = [X] 164.94/60.06 164.94/60.06 [U11(tt())] = [0] 164.94/60.06 [0] 164.94/60.06 >= [0] 164.94/60.06 [0] 164.94/60.06 = [U12(tt())] 164.94/60.06 164.94/60.06 [U12(tt())] = [0] 164.94/60.06 [0] 164.94/60.06 >= [0] 164.94/60.06 [0] 164.94/60.06 = [tt()] 164.94/60.06 164.94/60.06 [isNePal(__(I, __(P, I)))] = [2 2] I + [1 2] P + [2] 164.94/60.06 [0 0] [0 0] [2] 164.94/60.06 > [0] 164.94/60.06 [0] 164.94/60.06 = [U11(tt())] 164.94/60.06 164.94/60.06 [activate(X)] = [1 1] X + [1] 164.94/60.06 [0 1] [1] 164.94/60.06 > [1 0] X + [0] 164.94/60.06 [0 1] [0] 164.94/60.06 = [X] 164.94/60.06 164.94/60.06 164.94/60.06 We return to the main proof. 164.94/60.06 164.94/60.06 We are left with following problem, upon which TcT provides the 164.94/60.06 certificate YES(O(1),O(1)). 164.94/60.06 164.94/60.06 Weak Trs: 164.94/60.06 { __(X, nil()) -> X 164.94/60.06 , __(__(X, Y), Z) -> __(X, __(Y, Z)) 164.94/60.06 , __(nil(), X) -> X 164.94/60.06 , U11(tt()) -> U12(tt()) 164.94/60.06 , U12(tt()) -> tt() 164.94/60.06 , isNePal(__(I, __(P, I))) -> U11(tt()) 164.94/60.06 , activate(X) -> X } 164.94/60.06 Obligation: 164.94/60.06 derivational complexity 164.94/60.06 Answer: 164.94/60.06 YES(O(1),O(1)) 164.94/60.06 164.94/60.06 Empty rules are trivially bounded 164.94/60.06 164.94/60.06 Hurray, we answered YES(O(1),O(n^2)) 164.94/60.09 EOF