YES(O(1),O(n^3)) 178.74/60.08 YES(O(1),O(n^3)) 178.74/60.08 178.74/60.08 We are left with following problem, upon which TcT provides the 178.74/60.08 certificate YES(O(1),O(n^3)). 178.74/60.08 178.74/60.08 Strict Trs: 178.74/60.08 { 3(1(x1)) -> 4(1(x1)) 178.74/60.08 , 3(5(x1)) -> 8(9(7(x1))) 178.74/60.08 , 3(9(x1)) -> 9(3(x1)) 178.74/60.08 , 3(8(x1)) -> 3(2(7(x1))) 178.74/60.08 , 5(9(x1)) -> 2(6(5(x1))) 178.74/60.08 , 9(x1) -> 3(2(3(x1))) 178.74/60.08 , 9(x1) -> 5(0(2(x1))) 178.74/60.08 , 2(6(x1)) -> 4(3(x1)) 178.74/60.08 , 8(4(x1)) -> 6(x1) 178.74/60.08 , 8(8(4(x1))) -> 1(9(x1)) 178.74/60.08 , 7(1(x1)) -> 6(9(x1)) 178.74/60.08 , 7(5(x1)) -> 1(0(x1)) } 178.74/60.08 Obligation: 178.74/60.08 derivational complexity 178.74/60.08 Answer: 178.74/60.08 YES(O(1),O(n^3)) 178.74/60.08 178.74/60.08 The weightgap principle applies (using the following nonconstant 178.74/60.08 growth matrix-interpretation) 178.74/60.08 178.74/60.08 TcT has computed the following triangular matrix interpretation. 178.74/60.08 Note that the diagonal of the component-wise maxima of 178.74/60.08 interpretation-entries contains no more than 1 non-zero entries. 178.74/60.08 178.74/60.08 [3](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [1](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [4](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [5](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [9](x1) = [1] x1 + [1] 178.74/60.08 178.74/60.08 [2](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [6](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [8](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [7](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [0](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 The order satisfies the following ordering constraints: 178.74/60.08 178.74/60.08 [3(1(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [4(1(x1))] 178.74/60.08 178.74/60.08 [3(5(x1))] = [1] x1 + [0] 178.74/60.08 ? [1] x1 + [1] 178.74/60.08 = [8(9(7(x1)))] 178.74/60.08 178.74/60.08 [3(9(x1))] = [1] x1 + [1] 178.74/60.08 >= [1] x1 + [1] 178.74/60.08 = [9(3(x1))] 178.74/60.08 178.74/60.08 [3(8(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [3(2(7(x1)))] 178.74/60.08 178.74/60.08 [5(9(x1))] = [1] x1 + [1] 178.74/60.08 > [1] x1 + [0] 178.74/60.08 = [2(6(5(x1)))] 178.74/60.08 178.74/60.08 [9(x1)] = [1] x1 + [1] 178.74/60.08 > [1] x1 + [0] 178.74/60.08 = [3(2(3(x1)))] 178.74/60.08 178.74/60.08 [9(x1)] = [1] x1 + [1] 178.74/60.08 > [1] x1 + [0] 178.74/60.08 = [5(0(2(x1)))] 178.74/60.08 178.74/60.08 [2(6(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [4(3(x1))] 178.74/60.08 178.74/60.08 [8(4(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [6(x1)] 178.74/60.08 178.74/60.08 [8(8(4(x1)))] = [1] x1 + [0] 178.74/60.08 ? [1] x1 + [1] 178.74/60.08 = [1(9(x1))] 178.74/60.08 178.74/60.08 [7(1(x1))] = [1] x1 + [0] 178.74/60.08 ? [1] x1 + [1] 178.74/60.08 = [6(9(x1))] 178.74/60.08 178.74/60.08 [7(5(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [1(0(x1))] 178.74/60.08 178.74/60.08 178.74/60.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 178.74/60.08 178.74/60.08 We are left with following problem, upon which TcT provides the 178.74/60.08 certificate YES(O(1),O(n^3)). 178.74/60.08 178.74/60.08 Strict Trs: 178.74/60.08 { 3(1(x1)) -> 4(1(x1)) 178.74/60.08 , 3(5(x1)) -> 8(9(7(x1))) 178.74/60.08 , 3(9(x1)) -> 9(3(x1)) 178.74/60.08 , 3(8(x1)) -> 3(2(7(x1))) 178.74/60.08 , 2(6(x1)) -> 4(3(x1)) 178.74/60.08 , 8(4(x1)) -> 6(x1) 178.74/60.08 , 8(8(4(x1))) -> 1(9(x1)) 178.74/60.08 , 7(1(x1)) -> 6(9(x1)) 178.74/60.08 , 7(5(x1)) -> 1(0(x1)) } 178.74/60.08 Weak Trs: 178.74/60.08 { 5(9(x1)) -> 2(6(5(x1))) 178.74/60.08 , 9(x1) -> 3(2(3(x1))) 178.74/60.08 , 9(x1) -> 5(0(2(x1))) } 178.74/60.08 Obligation: 178.74/60.08 derivational complexity 178.74/60.08 Answer: 178.74/60.08 YES(O(1),O(n^3)) 178.74/60.08 178.74/60.08 The weightgap principle applies (using the following nonconstant 178.74/60.08 growth matrix-interpretation) 178.74/60.08 178.74/60.08 TcT has computed the following triangular matrix interpretation. 178.74/60.08 Note that the diagonal of the component-wise maxima of 178.74/60.08 interpretation-entries contains no more than 1 non-zero entries. 178.74/60.08 178.74/60.08 [3](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [1](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [4](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [5](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [9](x1) = [1] x1 + [1] 178.74/60.08 178.74/60.08 [2](x1) = [1] x1 + [1] 178.74/60.08 178.74/60.08 [6](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [8](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [7](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [0](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 The order satisfies the following ordering constraints: 178.74/60.08 178.74/60.08 [3(1(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [4(1(x1))] 178.74/60.08 178.74/60.08 [3(5(x1))] = [1] x1 + [0] 178.74/60.08 ? [1] x1 + [1] 178.74/60.08 = [8(9(7(x1)))] 178.74/60.08 178.74/60.08 [3(9(x1))] = [1] x1 + [1] 178.74/60.08 >= [1] x1 + [1] 178.74/60.08 = [9(3(x1))] 178.74/60.08 178.74/60.08 [3(8(x1))] = [1] x1 + [0] 178.74/60.08 ? [1] x1 + [1] 178.74/60.08 = [3(2(7(x1)))] 178.74/60.08 178.74/60.08 [5(9(x1))] = [1] x1 + [1] 178.74/60.08 >= [1] x1 + [1] 178.74/60.08 = [2(6(5(x1)))] 178.74/60.08 178.74/60.08 [9(x1)] = [1] x1 + [1] 178.74/60.08 >= [1] x1 + [1] 178.74/60.08 = [3(2(3(x1)))] 178.74/60.08 178.74/60.08 [9(x1)] = [1] x1 + [1] 178.74/60.08 >= [1] x1 + [1] 178.74/60.08 = [5(0(2(x1)))] 178.74/60.08 178.74/60.08 [2(6(x1))] = [1] x1 + [1] 178.74/60.08 > [1] x1 + [0] 178.74/60.08 = [4(3(x1))] 178.74/60.08 178.74/60.08 [8(4(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [6(x1)] 178.74/60.08 178.74/60.08 [8(8(4(x1)))] = [1] x1 + [0] 178.74/60.08 ? [1] x1 + [1] 178.74/60.08 = [1(9(x1))] 178.74/60.08 178.74/60.08 [7(1(x1))] = [1] x1 + [0] 178.74/60.08 ? [1] x1 + [1] 178.74/60.08 = [6(9(x1))] 178.74/60.08 178.74/60.08 [7(5(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [1(0(x1))] 178.74/60.08 178.74/60.08 178.74/60.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 178.74/60.08 178.74/60.08 We are left with following problem, upon which TcT provides the 178.74/60.08 certificate YES(O(1),O(n^3)). 178.74/60.08 178.74/60.08 Strict Trs: 178.74/60.08 { 3(1(x1)) -> 4(1(x1)) 178.74/60.08 , 3(5(x1)) -> 8(9(7(x1))) 178.74/60.08 , 3(9(x1)) -> 9(3(x1)) 178.74/60.08 , 3(8(x1)) -> 3(2(7(x1))) 178.74/60.08 , 8(4(x1)) -> 6(x1) 178.74/60.08 , 8(8(4(x1))) -> 1(9(x1)) 178.74/60.08 , 7(1(x1)) -> 6(9(x1)) 178.74/60.08 , 7(5(x1)) -> 1(0(x1)) } 178.74/60.08 Weak Trs: 178.74/60.08 { 5(9(x1)) -> 2(6(5(x1))) 178.74/60.08 , 9(x1) -> 3(2(3(x1))) 178.74/60.08 , 9(x1) -> 5(0(2(x1))) 178.74/60.08 , 2(6(x1)) -> 4(3(x1)) } 178.74/60.08 Obligation: 178.74/60.08 derivational complexity 178.74/60.08 Answer: 178.74/60.08 YES(O(1),O(n^3)) 178.74/60.08 178.74/60.08 The weightgap principle applies (using the following nonconstant 178.74/60.08 growth matrix-interpretation) 178.74/60.08 178.74/60.08 TcT has computed the following triangular matrix interpretation. 178.74/60.08 Note that the diagonal of the component-wise maxima of 178.74/60.08 interpretation-entries contains no more than 1 non-zero entries. 178.74/60.08 178.74/60.08 [3](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [1](x1) = [1] x1 + [1] 178.74/60.08 178.74/60.08 [4](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [5](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [9](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [2](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [6](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [8](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [7](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 [0](x1) = [1] x1 + [0] 178.74/60.08 178.74/60.08 The order satisfies the following ordering constraints: 178.74/60.08 178.74/60.08 [3(1(x1))] = [1] x1 + [1] 178.74/60.08 >= [1] x1 + [1] 178.74/60.08 = [4(1(x1))] 178.74/60.08 178.74/60.08 [3(5(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [8(9(7(x1)))] 178.74/60.08 178.74/60.08 [3(9(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [9(3(x1))] 178.74/60.08 178.74/60.08 [3(8(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [3(2(7(x1)))] 178.74/60.08 178.74/60.08 [5(9(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [2(6(5(x1)))] 178.74/60.08 178.74/60.08 [9(x1)] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [3(2(3(x1)))] 178.74/60.08 178.74/60.08 [9(x1)] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [5(0(2(x1)))] 178.74/60.08 178.74/60.08 [2(6(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [4(3(x1))] 178.74/60.08 178.74/60.08 [8(4(x1))] = [1] x1 + [0] 178.74/60.08 >= [1] x1 + [0] 178.74/60.08 = [6(x1)] 178.74/60.08 178.74/60.08 [8(8(4(x1)))] = [1] x1 + [0] 178.74/60.08 ? [1] x1 + [1] 178.74/60.08 = [1(9(x1))] 178.74/60.08 178.74/60.08 [7(1(x1))] = [1] x1 + [1] 178.74/60.08 > [1] x1 + [0] 178.74/60.08 = [6(9(x1))] 178.74/60.08 178.74/60.08 [7(5(x1))] = [1] x1 + [0] 178.74/60.08 ? [1] x1 + [1] 178.74/60.08 = [1(0(x1))] 178.74/60.08 178.74/60.08 178.74/60.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 178.74/60.08 178.74/60.08 We are left with following problem, upon which TcT provides the 178.74/60.08 certificate YES(O(1),O(n^3)). 178.74/60.08 178.74/60.08 Strict Trs: 178.74/60.08 { 3(1(x1)) -> 4(1(x1)) 178.74/60.08 , 3(5(x1)) -> 8(9(7(x1))) 178.74/60.09 , 3(9(x1)) -> 9(3(x1)) 178.74/60.09 , 3(8(x1)) -> 3(2(7(x1))) 178.74/60.09 , 8(4(x1)) -> 6(x1) 178.74/60.09 , 8(8(4(x1))) -> 1(9(x1)) 178.74/60.09 , 7(5(x1)) -> 1(0(x1)) } 178.74/60.09 Weak Trs: 178.74/60.09 { 5(9(x1)) -> 2(6(5(x1))) 178.74/60.09 , 9(x1) -> 3(2(3(x1))) 178.74/60.09 , 9(x1) -> 5(0(2(x1))) 178.74/60.09 , 2(6(x1)) -> 4(3(x1)) 178.74/60.09 , 7(1(x1)) -> 6(9(x1)) } 178.74/60.09 Obligation: 178.74/60.09 derivational complexity 178.74/60.09 Answer: 178.74/60.09 YES(O(1),O(n^3)) 178.74/60.09 178.74/60.09 The weightgap principle applies (using the following nonconstant 178.74/60.09 growth matrix-interpretation) 178.74/60.09 178.74/60.09 TcT has computed the following triangular matrix interpretation. 178.74/60.09 Note that the diagonal of the component-wise maxima of 178.74/60.09 interpretation-entries contains no more than 1 non-zero entries. 178.74/60.09 178.74/60.09 [3](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [1](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [4](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [5](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [9](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [2](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [6](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [8](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [7](x1) = [1] x1 + [1] 178.74/60.09 178.74/60.09 [0](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 The order satisfies the following ordering constraints: 178.74/60.09 178.74/60.09 [3(1(x1))] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [4(1(x1))] 178.74/60.09 178.74/60.09 [3(5(x1))] = [1] x1 + [0] 178.74/60.09 ? [1] x1 + [1] 178.74/60.09 = [8(9(7(x1)))] 178.74/60.09 178.74/60.09 [3(9(x1))] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [9(3(x1))] 178.74/60.09 178.74/60.09 [3(8(x1))] = [1] x1 + [0] 178.74/60.09 ? [1] x1 + [1] 178.74/60.09 = [3(2(7(x1)))] 178.74/60.09 178.74/60.09 [5(9(x1))] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [2(6(5(x1)))] 178.74/60.09 178.74/60.09 [9(x1)] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [3(2(3(x1)))] 178.74/60.09 178.74/60.09 [9(x1)] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [5(0(2(x1)))] 178.74/60.09 178.74/60.09 [2(6(x1))] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [4(3(x1))] 178.74/60.09 178.74/60.09 [8(4(x1))] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [6(x1)] 178.74/60.09 178.74/60.09 [8(8(4(x1)))] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [1(9(x1))] 178.74/60.09 178.74/60.09 [7(1(x1))] = [1] x1 + [1] 178.74/60.09 > [1] x1 + [0] 178.74/60.09 = [6(9(x1))] 178.74/60.09 178.74/60.09 [7(5(x1))] = [1] x1 + [1] 178.74/60.09 > [1] x1 + [0] 178.74/60.09 = [1(0(x1))] 178.74/60.09 178.74/60.09 178.74/60.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 178.74/60.09 178.74/60.09 We are left with following problem, upon which TcT provides the 178.74/60.09 certificate YES(O(1),O(n^3)). 178.74/60.09 178.74/60.09 Strict Trs: 178.74/60.09 { 3(1(x1)) -> 4(1(x1)) 178.74/60.09 , 3(5(x1)) -> 8(9(7(x1))) 178.74/60.09 , 3(9(x1)) -> 9(3(x1)) 178.74/60.09 , 3(8(x1)) -> 3(2(7(x1))) 178.74/60.09 , 8(4(x1)) -> 6(x1) 178.74/60.09 , 8(8(4(x1))) -> 1(9(x1)) } 178.74/60.09 Weak Trs: 178.74/60.09 { 5(9(x1)) -> 2(6(5(x1))) 178.74/60.09 , 9(x1) -> 3(2(3(x1))) 178.74/60.09 , 9(x1) -> 5(0(2(x1))) 178.74/60.09 , 2(6(x1)) -> 4(3(x1)) 178.74/60.09 , 7(1(x1)) -> 6(9(x1)) 178.74/60.09 , 7(5(x1)) -> 1(0(x1)) } 178.74/60.09 Obligation: 178.74/60.09 derivational complexity 178.74/60.09 Answer: 178.74/60.09 YES(O(1),O(n^3)) 178.74/60.09 178.74/60.09 The weightgap principle applies (using the following nonconstant 178.74/60.09 growth matrix-interpretation) 178.74/60.09 178.74/60.09 TcT has computed the following triangular matrix interpretation. 178.74/60.09 Note that the diagonal of the component-wise maxima of 178.74/60.09 interpretation-entries contains no more than 1 non-zero entries. 178.74/60.09 178.74/60.09 [3](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [1](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [4](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [5](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [9](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [2](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [6](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [8](x1) = [1] x1 + [1] 178.74/60.09 178.74/60.09 [7](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [0](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 The order satisfies the following ordering constraints: 178.74/60.09 178.74/60.09 [3(1(x1))] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [4(1(x1))] 178.74/60.09 178.74/60.09 [3(5(x1))] = [1] x1 + [0] 178.74/60.09 ? [1] x1 + [1] 178.74/60.09 = [8(9(7(x1)))] 178.74/60.09 178.74/60.09 [3(9(x1))] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [9(3(x1))] 178.74/60.09 178.74/60.09 [3(8(x1))] = [1] x1 + [1] 178.74/60.09 > [1] x1 + [0] 178.74/60.09 = [3(2(7(x1)))] 178.74/60.09 178.74/60.09 [5(9(x1))] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [2(6(5(x1)))] 178.74/60.09 178.74/60.09 [9(x1)] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [3(2(3(x1)))] 178.74/60.09 178.74/60.09 [9(x1)] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [5(0(2(x1)))] 178.74/60.09 178.74/60.09 [2(6(x1))] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [4(3(x1))] 178.74/60.09 178.74/60.09 [8(4(x1))] = [1] x1 + [1] 178.74/60.09 > [1] x1 + [0] 178.74/60.09 = [6(x1)] 178.74/60.09 178.74/60.09 [8(8(4(x1)))] = [1] x1 + [2] 178.74/60.09 > [1] x1 + [0] 178.74/60.09 = [1(9(x1))] 178.74/60.09 178.74/60.09 [7(1(x1))] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [6(9(x1))] 178.74/60.09 178.74/60.09 [7(5(x1))] = [1] x1 + [0] 178.74/60.09 >= [1] x1 + [0] 178.74/60.09 = [1(0(x1))] 178.74/60.09 178.74/60.09 178.74/60.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 178.74/60.09 178.74/60.09 We are left with following problem, upon which TcT provides the 178.74/60.09 certificate YES(O(1),O(n^3)). 178.74/60.09 178.74/60.09 Strict Trs: 178.74/60.09 { 3(1(x1)) -> 4(1(x1)) 178.74/60.09 , 3(5(x1)) -> 8(9(7(x1))) 178.74/60.09 , 3(9(x1)) -> 9(3(x1)) } 178.74/60.09 Weak Trs: 178.74/60.09 { 3(8(x1)) -> 3(2(7(x1))) 178.74/60.09 , 5(9(x1)) -> 2(6(5(x1))) 178.74/60.09 , 9(x1) -> 3(2(3(x1))) 178.74/60.09 , 9(x1) -> 5(0(2(x1))) 178.74/60.09 , 2(6(x1)) -> 4(3(x1)) 178.74/60.09 , 8(4(x1)) -> 6(x1) 178.74/60.09 , 8(8(4(x1))) -> 1(9(x1)) 178.74/60.09 , 7(1(x1)) -> 6(9(x1)) 178.74/60.09 , 7(5(x1)) -> 1(0(x1)) } 178.74/60.09 Obligation: 178.74/60.09 derivational complexity 178.74/60.09 Answer: 178.74/60.09 YES(O(1),O(n^3)) 178.74/60.09 178.74/60.09 The weightgap principle applies (using the following nonconstant 178.74/60.09 growth matrix-interpretation) 178.74/60.09 178.74/60.09 TcT has computed the following triangular matrix interpretation. 178.74/60.09 Note that the diagonal of the component-wise maxima of 178.74/60.09 interpretation-entries contains no more than 1 non-zero entries. 178.74/60.09 178.74/60.09 [3](x1) = [1] x1 + [1] 178.74/60.09 178.74/60.09 [1](x1) = [1] x1 + [1] 178.74/60.09 178.74/60.09 [4](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [5](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [9](x1) = [1] x1 + [2] 178.74/60.09 178.74/60.09 [2](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 [6](x1) = [1] x1 + [1] 178.74/60.09 178.74/60.09 [8](x1) = [1] x1 + [2] 178.74/60.09 178.74/60.09 [7](x1) = [1] x1 + [2] 178.74/60.09 178.74/60.09 [0](x1) = [1] x1 + [0] 178.74/60.09 178.74/60.09 The order satisfies the following ordering constraints: 178.74/60.09 178.74/60.09 [3(1(x1))] = [1] x1 + [2] 178.74/60.09 > [1] x1 + [1] 178.74/60.09 = [4(1(x1))] 178.74/60.09 178.74/60.09 [3(5(x1))] = [1] x1 + [1] 178.74/60.09 ? [1] x1 + [6] 178.74/60.09 = [8(9(7(x1)))] 178.74/60.09 178.74/60.09 [3(9(x1))] = [1] x1 + [3] 178.74/60.09 >= [1] x1 + [3] 178.74/60.09 = [9(3(x1))] 178.74/60.09 178.74/60.09 [3(8(x1))] = [1] x1 + [3] 178.74/60.09 >= [1] x1 + [3] 178.74/60.09 = [3(2(7(x1)))] 178.74/60.09 178.74/60.09 [5(9(x1))] = [1] x1 + [2] 178.74/60.09 > [1] x1 + [1] 178.74/60.09 = [2(6(5(x1)))] 178.74/60.09 178.74/60.09 [9(x1)] = [1] x1 + [2] 178.74/60.09 >= [1] x1 + [2] 178.74/60.09 = [3(2(3(x1)))] 178.74/60.09 178.74/60.09 [9(x1)] = [1] x1 + [2] 178.74/60.09 > [1] x1 + [0] 178.74/60.09 = [5(0(2(x1)))] 178.74/60.09 178.74/60.09 [2(6(x1))] = [1] x1 + [1] 178.74/60.09 >= [1] x1 + [1] 178.74/60.09 = [4(3(x1))] 178.74/60.09 178.74/60.09 [8(4(x1))] = [1] x1 + [2] 178.74/60.09 > [1] x1 + [1] 178.74/60.09 = [6(x1)] 178.74/60.09 178.74/60.09 [8(8(4(x1)))] = [1] x1 + [4] 178.74/60.09 > [1] x1 + [3] 178.74/60.09 = [1(9(x1))] 178.74/60.09 178.74/60.09 [7(1(x1))] = [1] x1 + [3] 178.74/60.09 >= [1] x1 + [3] 178.74/60.09 = [6(9(x1))] 178.74/60.09 178.74/60.09 [7(5(x1))] = [1] x1 + [2] 178.74/60.09 > [1] x1 + [1] 178.74/60.09 = [1(0(x1))] 178.74/60.09 178.74/60.09 178.74/60.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 178.74/60.09 178.74/60.09 We are left with following problem, upon which TcT provides the 178.74/60.09 certificate YES(O(1),O(n^3)). 178.74/60.09 178.74/60.09 Strict Trs: 178.74/60.09 { 3(5(x1)) -> 8(9(7(x1))) 178.74/60.09 , 3(9(x1)) -> 9(3(x1)) } 178.74/60.09 Weak Trs: 178.74/60.09 { 3(1(x1)) -> 4(1(x1)) 178.74/60.09 , 3(8(x1)) -> 3(2(7(x1))) 178.74/60.09 , 5(9(x1)) -> 2(6(5(x1))) 178.74/60.09 , 9(x1) -> 3(2(3(x1))) 178.74/60.09 , 9(x1) -> 5(0(2(x1))) 178.74/60.09 , 2(6(x1)) -> 4(3(x1)) 178.74/60.09 , 8(4(x1)) -> 6(x1) 178.74/60.09 , 8(8(4(x1))) -> 1(9(x1)) 178.74/60.09 , 7(1(x1)) -> 6(9(x1)) 178.74/60.09 , 7(5(x1)) -> 1(0(x1)) } 178.74/60.09 Obligation: 178.74/60.09 derivational complexity 178.74/60.09 Answer: 178.74/60.09 YES(O(1),O(n^3)) 178.74/60.09 178.74/60.09 We use the processor 'matrix interpretation of dimension 3' to 178.74/60.09 orient following rules strictly. 178.74/60.09 178.74/60.09 Trs: { 3(5(x1)) -> 8(9(7(x1))) } 178.74/60.09 178.74/60.09 The induced complexity on above rules (modulo remaining rules) is 178.74/60.09 YES(?,O(n^1)) . These rules are moved into the corresponding weak 178.74/60.09 component(s). 178.74/60.09 178.74/60.09 Sub-proof: 178.74/60.09 ---------- 178.74/60.09 TcT has computed the following triangular matrix interpretation. 178.74/60.09 Note that the diagonal of the component-wise maxima of 178.74/60.09 interpretation-entries contains no more than 1 non-zero entries. 178.74/60.09 178.74/60.09 [1 0 2] [0] 178.74/60.09 [3](x1) = [0 0 2] x1 + [0] 178.74/60.09 [0 0 0] [1] 178.74/60.09 178.74/60.09 [1 0 2] [1] 178.74/60.09 [1](x1) = [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 178.74/60.09 [1 0 0] [0] 178.74/60.09 [4](x1) = [0 0 1] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 178.74/60.09 [1 0 2] [0] 178.74/60.09 [5](x1) = [0 0 1] x1 + [0] 178.74/60.09 [0 0 0] [1] 178.74/60.09 178.74/60.09 [1 0 2] [0] 178.74/60.09 [9](x1) = [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [1] 178.74/60.09 178.74/60.09 [1 0 0] [0] 178.74/60.09 [2](x1) = [0 0 0] x1 + [1] 178.74/60.09 [0 0 0] [0] 178.74/60.09 178.74/60.09 [1 0 2] [0] 178.74/60.09 [6](x1) = [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 178.74/60.09 [1 2 0] [0] 178.74/60.09 [8](x1) = [0 0 0] x1 + [2] 178.74/60.09 [0 0 0] [1] 178.74/60.09 178.74/60.09 [1 0 0] [1] 178.74/60.09 [7](x1) = [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 178.74/60.09 [1 0 0] [0] 178.74/60.09 [0](x1) = [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 178.74/60.09 The order satisfies the following ordering constraints: 178.74/60.09 178.74/60.09 [3(1(x1))] = [1 0 2] [1] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [1] 178.74/60.09 >= [1 0 2] [1] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [4(1(x1))] 178.74/60.09 178.74/60.09 [3(5(x1))] = [1 0 2] [2] 178.74/60.09 [0 0 0] x1 + [2] 178.74/60.09 [0 0 0] [1] 178.74/60.09 > [1 0 0] [1] 178.74/60.09 [0 0 0] x1 + [2] 178.74/60.09 [0 0 0] [1] 178.74/60.09 = [8(9(7(x1)))] 178.74/60.09 178.74/60.09 [3(9(x1))] = [1 0 2] [2] 178.74/60.09 [0 0 0] x1 + [2] 178.74/60.09 [0 0 0] [1] 178.74/60.09 >= [1 0 2] [2] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [1] 178.74/60.09 = [9(3(x1))] 178.74/60.09 178.74/60.09 [3(8(x1))] = [1 2 0] [2] 178.74/60.09 [0 0 0] x1 + [2] 178.74/60.09 [0 0 0] [1] 178.74/60.09 > [1 0 0] [1] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [1] 178.74/60.09 = [3(2(7(x1)))] 178.74/60.09 178.74/60.09 [5(9(x1))] = [1 0 2] [2] 178.74/60.09 [0 0 0] x1 + [1] 178.74/60.09 [0 0 0] [1] 178.74/60.09 >= [1 0 2] [2] 178.74/60.09 [0 0 0] x1 + [1] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [2(6(5(x1)))] 178.74/60.09 178.74/60.09 [9(x1)] = [1 0 2] [0] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [1] 178.74/60.09 >= [1 0 2] [0] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [1] 178.74/60.09 = [3(2(3(x1)))] 178.74/60.09 178.74/60.09 [9(x1)] = [1 0 2] [0] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [1] 178.74/60.09 >= [1 0 0] [0] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [1] 178.74/60.09 = [5(0(2(x1)))] 178.74/60.09 178.74/60.09 [2(6(x1))] = [1 0 2] [0] 178.74/60.09 [0 0 0] x1 + [1] 178.74/60.09 [0 0 0] [0] 178.74/60.09 >= [1 0 2] [0] 178.74/60.09 [0 0 0] x1 + [1] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [4(3(x1))] 178.74/60.09 178.74/60.09 [8(4(x1))] = [1 0 2] [0] 178.74/60.09 [0 0 0] x1 + [2] 178.74/60.09 [0 0 0] [1] 178.74/60.09 >= [1 0 2] [0] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [6(x1)] 178.74/60.09 178.74/60.09 [8(8(4(x1)))] = [1 0 2] [4] 178.74/60.09 [0 0 0] x1 + [2] 178.74/60.09 [0 0 0] [1] 178.74/60.09 > [1 0 2] [3] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [1(9(x1))] 178.74/60.09 178.74/60.09 [7(1(x1))] = [1 0 2] [2] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 >= [1 0 2] [2] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [6(9(x1))] 178.74/60.09 178.74/60.09 [7(5(x1))] = [1 0 2] [1] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 >= [1 0 0] [1] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [1(0(x1))] 178.74/60.09 178.74/60.09 178.74/60.09 We return to the main proof. 178.74/60.09 178.74/60.09 We are left with following problem, upon which TcT provides the 178.74/60.09 certificate YES(O(1),O(n^3)). 178.74/60.09 178.74/60.09 Strict Trs: { 3(9(x1)) -> 9(3(x1)) } 178.74/60.09 Weak Trs: 178.74/60.09 { 3(1(x1)) -> 4(1(x1)) 178.74/60.09 , 3(5(x1)) -> 8(9(7(x1))) 178.74/60.09 , 3(8(x1)) -> 3(2(7(x1))) 178.74/60.09 , 5(9(x1)) -> 2(6(5(x1))) 178.74/60.09 , 9(x1) -> 3(2(3(x1))) 178.74/60.09 , 9(x1) -> 5(0(2(x1))) 178.74/60.09 , 2(6(x1)) -> 4(3(x1)) 178.74/60.09 , 8(4(x1)) -> 6(x1) 178.74/60.09 , 8(8(4(x1))) -> 1(9(x1)) 178.74/60.09 , 7(1(x1)) -> 6(9(x1)) 178.74/60.09 , 7(5(x1)) -> 1(0(x1)) } 178.74/60.09 Obligation: 178.74/60.09 derivational complexity 178.74/60.09 Answer: 178.74/60.09 YES(O(1),O(n^3)) 178.74/60.09 178.74/60.09 We use the processor 'matrix interpretation of dimension 3' to 178.74/60.09 orient following rules strictly. 178.74/60.09 178.74/60.09 Trs: { 3(9(x1)) -> 9(3(x1)) } 178.74/60.09 178.74/60.09 The induced complexity on above rules (modulo remaining rules) is 178.74/60.09 YES(?,O(n^3)) . These rules are moved into the corresponding weak 178.74/60.09 component(s). 178.74/60.09 178.74/60.09 Sub-proof: 178.74/60.09 ---------- 178.74/60.09 TcT has computed the following triangular matrix interpretation. 178.74/60.09 178.74/60.09 [1 0 1] [0] 178.74/60.09 [3](x1) = [0 1 2] x1 + [0] 178.74/60.09 [0 0 1] [0] 178.74/60.09 178.74/60.09 [1 0 1] [0] 178.74/60.09 [1](x1) = [0 0 2] x1 + [2] 178.74/60.09 [0 0 0] [0] 178.74/60.09 178.74/60.09 [1 0 0] [0] 178.74/60.09 [4](x1) = [0 0 2] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 178.74/60.09 [1 2 1] [0] 178.74/60.09 [5](x1) = [0 1 2] x1 + [0] 178.74/60.09 [0 0 1] [1] 178.74/60.09 178.74/60.09 [1 0 2] [0] 178.74/60.09 [9](x1) = [0 1 2] x1 + [0] 178.74/60.09 [0 0 1] [1] 178.74/60.09 178.74/60.09 [1 0 0] [0] 178.74/60.09 [2](x1) = [0 1 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 178.74/60.09 [1 0 1] [0] 178.74/60.09 [6](x1) = [0 0 2] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 178.74/60.09 [1 1 0] [0] 178.74/60.09 [8](x1) = [0 1 0] x1 + [2] 178.74/60.09 [0 0 1] [0] 178.74/60.09 178.74/60.09 [1 1 0] [0] 178.74/60.09 [7](x1) = [0 1 2] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 178.74/60.09 [1 0 0] [0] 178.74/60.09 [0](x1) = [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 178.74/60.09 The order satisfies the following ordering constraints: 178.74/60.09 178.74/60.09 [3(1(x1))] = [1 0 1] [0] 178.74/60.09 [0 0 2] x1 + [2] 178.74/60.09 [0 0 0] [0] 178.74/60.09 >= [1 0 1] [0] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [4(1(x1))] 178.74/60.09 178.74/60.09 [3(5(x1))] = [1 2 2] [1] 178.74/60.09 [0 1 4] x1 + [2] 178.74/60.09 [0 0 1] [1] 178.74/60.09 > [1 2 2] [0] 178.74/60.09 [0 1 2] x1 + [2] 178.74/60.09 [0 0 0] [1] 178.74/60.09 = [8(9(7(x1)))] 178.74/60.09 178.74/60.09 [3(9(x1))] = [1 0 3] [1] 178.74/60.09 [0 1 4] x1 + [2] 178.74/60.09 [0 0 1] [1] 178.74/60.09 > [1 0 3] [0] 178.74/60.09 [0 1 4] x1 + [0] 178.74/60.09 [0 0 1] [1] 178.74/60.09 = [9(3(x1))] 178.74/60.09 178.74/60.09 [3(8(x1))] = [1 1 1] [0] 178.74/60.09 [0 1 2] x1 + [2] 178.74/60.09 [0 0 1] [0] 178.74/60.09 >= [1 1 0] [0] 178.74/60.09 [0 1 2] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [3(2(7(x1)))] 178.74/60.09 178.74/60.09 [5(9(x1))] = [1 2 7] [1] 178.74/60.09 [0 1 4] x1 + [2] 178.74/60.09 [0 0 1] [2] 178.74/60.09 >= [1 2 2] [1] 178.74/60.09 [0 0 2] x1 + [2] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [2(6(5(x1)))] 178.74/60.09 178.74/60.09 [9(x1)] = [1 0 2] [0] 178.74/60.09 [0 1 2] x1 + [0] 178.74/60.09 [0 0 1] [1] 178.74/60.09 >= [1 0 1] [0] 178.74/60.09 [0 1 2] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [3(2(3(x1)))] 178.74/60.09 178.74/60.09 [9(x1)] = [1 0 2] [0] 178.74/60.09 [0 1 2] x1 + [0] 178.74/60.09 [0 0 1] [1] 178.74/60.09 >= [1 0 0] [0] 178.74/60.09 [0 0 0] x1 + [0] 178.74/60.09 [0 0 0] [1] 178.74/60.09 = [5(0(2(x1)))] 178.74/60.09 178.74/60.09 [2(6(x1))] = [1 0 1] [0] 178.74/60.09 [0 0 2] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 >= [1 0 1] [0] 178.74/60.09 [0 0 2] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [4(3(x1))] 178.74/60.09 178.74/60.09 [8(4(x1))] = [1 0 2] [0] 178.74/60.09 [0 0 2] x1 + [2] 178.74/60.09 [0 0 0] [0] 178.74/60.09 >= [1 0 1] [0] 178.74/60.09 [0 0 2] x1 + [0] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [6(x1)] 178.74/60.09 178.74/60.09 [8(8(4(x1)))] = [1 0 4] [2] 178.74/60.09 [0 0 2] x1 + [4] 178.74/60.09 [0 0 0] [0] 178.74/60.09 > [1 0 3] [1] 178.74/60.09 [0 0 2] x1 + [4] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [1(9(x1))] 178.74/60.09 178.74/60.09 [7(1(x1))] = [1 0 3] [2] 178.74/60.09 [0 0 2] x1 + [2] 178.74/60.09 [0 0 0] [0] 178.74/60.09 > [1 0 3] [1] 178.74/60.09 [0 0 2] x1 + [2] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [6(9(x1))] 178.74/60.09 178.74/60.09 [7(5(x1))] = [1 3 3] [0] 178.74/60.09 [0 1 4] x1 + [2] 178.74/60.09 [0 0 0] [0] 178.74/60.09 >= [1 0 0] [0] 178.74/60.09 [0 0 0] x1 + [2] 178.74/60.09 [0 0 0] [0] 178.74/60.09 = [1(0(x1))] 178.74/60.09 178.74/60.09 178.74/60.09 We return to the main proof. 178.74/60.09 178.74/60.09 We are left with following problem, upon which TcT provides the 178.74/60.09 certificate YES(O(1),O(1)). 178.74/60.09 178.74/60.09 Weak Trs: 178.74/60.09 { 3(1(x1)) -> 4(1(x1)) 178.74/60.09 , 3(5(x1)) -> 8(9(7(x1))) 178.74/60.09 , 3(9(x1)) -> 9(3(x1)) 178.74/60.09 , 3(8(x1)) -> 3(2(7(x1))) 178.74/60.09 , 5(9(x1)) -> 2(6(5(x1))) 178.74/60.09 , 9(x1) -> 3(2(3(x1))) 178.74/60.09 , 9(x1) -> 5(0(2(x1))) 178.74/60.09 , 2(6(x1)) -> 4(3(x1)) 178.74/60.09 , 8(4(x1)) -> 6(x1) 178.74/60.09 , 8(8(4(x1))) -> 1(9(x1)) 178.74/60.09 , 7(1(x1)) -> 6(9(x1)) 178.74/60.09 , 7(5(x1)) -> 1(0(x1)) } 178.74/60.09 Obligation: 178.74/60.09 derivational complexity 178.74/60.09 Answer: 178.74/60.09 YES(O(1),O(1)) 178.74/60.09 178.74/60.09 Empty rules are trivially bounded 178.74/60.09 178.74/60.09 Hurray, we answered YES(O(1),O(n^3)) 178.74/60.09 EOF