MAYBE 205.19/60.05 MAYBE 205.19/60.05 205.19/60.05 We are left with following problem, upon which TcT provides the 205.19/60.05 certificate MAYBE. 205.19/60.05 205.19/60.05 Strict Trs: 205.19/60.05 { minus(minus(x)) -> x 205.19/60.05 , minus(0()) -> 0() 205.19/60.05 , +(x, minus(y)) -> minus(+(minus(x), y)) 205.19/60.05 , +(x, 0()) -> x 205.19/60.05 , +(x, +(y, z)) -> +(+(x, y), z) 205.19/60.05 , +(minus(+(x, 1())), 1()) -> minus(x) 205.19/60.05 , +(minus(1()), 1()) -> 0() 205.19/60.05 , +(0(), y) -> y } 205.19/60.05 Obligation: 205.19/60.05 derivational complexity 205.19/60.05 Answer: 205.19/60.05 MAYBE 205.19/60.05 205.19/60.05 None of the processors succeeded. 205.19/60.05 205.19/60.05 Details of failed attempt(s): 205.19/60.05 ----------------------------- 205.19/60.05 1) 'Fastest (timeout of 60 seconds)' failed due to the following 205.19/60.05 reason: 205.19/60.05 205.19/60.05 Computation stopped due to timeout after 60.0 seconds. 205.19/60.05 205.19/60.05 2) 'Inspecting Problem... (timeout of 297 seconds)' failed due to 205.19/60.05 the following reason: 205.19/60.05 205.19/60.05 We use the processor 'matrix interpretation of dimension 1' to 205.19/60.05 orient following rules strictly. 205.19/60.05 205.19/60.05 Trs: 205.19/60.05 { +(x, 0()) -> x 205.19/60.05 , +(minus(+(x, 1())), 1()) -> minus(x) 205.19/60.05 , +(minus(1()), 1()) -> 0() 205.19/60.05 , +(0(), y) -> y } 205.19/60.05 205.19/60.05 The induced complexity on above rules (modulo remaining rules) is 205.19/60.05 YES(?,O(n^1)) . These rules are moved into the corresponding weak 205.19/60.05 component(s). 205.19/60.05 205.19/60.05 Sub-proof: 205.19/60.05 ---------- 205.19/60.05 TcT has computed the following triangular matrix interpretation. 205.19/60.05 205.19/60.05 [minus](x1) = [1] x1 + [0] 205.19/60.05 205.19/60.05 [0] = [2] 205.19/60.05 205.19/60.05 [+](x1, x2) = [1] x1 + [1] x2 + [0] 205.19/60.05 205.19/60.05 [1] = [2] 205.19/60.05 205.19/60.05 The order satisfies the following ordering constraints: 205.19/60.05 205.19/60.05 [minus(minus(x))] = [1] x + [0] 205.19/60.05 >= [1] x + [0] 205.19/60.05 = [x] 205.19/60.05 205.19/60.05 [minus(0())] = [2] 205.19/60.05 >= [2] 205.19/60.05 = [0()] 205.19/60.05 205.19/60.05 [+(x, minus(y))] = [1] x + [1] y + [0] 205.19/60.05 >= [1] x + [1] y + [0] 205.19/60.05 = [minus(+(minus(x), y))] 205.19/60.05 205.19/60.05 [+(x, 0())] = [1] x + [2] 205.19/60.05 > [1] x + [0] 205.19/60.05 = [x] 205.19/60.05 205.19/60.05 [+(x, +(y, z))] = [1] x + [1] y + [1] z + [0] 205.19/60.05 >= [1] x + [1] y + [1] z + [0] 205.19/60.05 = [+(+(x, y), z)] 205.19/60.05 205.19/60.05 [+(minus(+(x, 1())), 1())] = [1] x + [4] 205.19/60.05 > [1] x + [0] 205.19/60.05 = [minus(x)] 205.19/60.05 205.19/60.05 [+(minus(1()), 1())] = [4] 205.19/60.05 > [2] 205.19/60.05 = [0()] 205.19/60.05 205.19/60.05 [+(0(), y)] = [1] y + [2] 205.19/60.05 > [1] y + [0] 205.19/60.05 = [y] 205.19/60.05 205.19/60.05 205.19/60.05 We return to the main proof. 205.19/60.05 205.19/60.05 We are left with following problem, upon which TcT provides the 205.19/60.05 certificate MAYBE. 205.19/60.05 205.19/60.05 Strict Trs: 205.19/60.05 { minus(minus(x)) -> x 205.19/60.05 , minus(0()) -> 0() 205.19/60.05 , +(x, minus(y)) -> minus(+(minus(x), y)) 205.19/60.05 , +(x, +(y, z)) -> +(+(x, y), z) } 205.19/60.05 Weak Trs: 205.19/60.05 { +(x, 0()) -> x 205.19/60.05 , +(minus(+(x, 1())), 1()) -> minus(x) 205.19/60.05 , +(minus(1()), 1()) -> 0() 205.19/60.05 , +(0(), y) -> y } 205.19/60.05 Obligation: 205.19/60.05 derivational complexity 205.19/60.05 Answer: 205.19/60.05 MAYBE 205.19/60.05 205.19/60.05 The weightgap principle applies (using the following nonconstant 205.19/60.05 growth matrix-interpretation) 205.19/60.05 205.19/60.05 TcT has computed the following triangular matrix interpretation. 205.19/60.05 Note that the diagonal of the component-wise maxima of 205.19/60.05 interpretation-entries contains no more than 1 non-zero entries. 205.19/60.05 205.19/60.05 [minus](x1) = [1] x1 + [1] 205.19/60.05 205.19/60.05 [0] = [1] 205.19/60.05 205.19/60.05 [+](x1, x2) = [1] x1 + [1] x2 + [0] 205.19/60.05 205.19/60.05 [1] = [0] 205.19/60.05 205.19/60.05 The order satisfies the following ordering constraints: 205.19/60.05 205.19/60.05 [minus(minus(x))] = [1] x + [2] 205.19/60.05 > [1] x + [0] 205.19/60.05 = [x] 205.19/60.05 205.19/60.05 [minus(0())] = [2] 205.19/60.05 > [1] 205.19/60.05 = [0()] 205.19/60.05 205.19/60.05 [+(x, minus(y))] = [1] x + [1] y + [1] 205.19/60.05 ? [1] x + [1] y + [2] 205.19/60.05 = [minus(+(minus(x), y))] 205.19/60.05 205.19/60.05 [+(x, 0())] = [1] x + [1] 205.19/60.05 > [1] x + [0] 205.19/60.05 = [x] 205.19/60.05 205.19/60.05 [+(x, +(y, z))] = [1] x + [1] y + [1] z + [0] 205.19/60.05 >= [1] x + [1] y + [1] z + [0] 205.19/60.05 = [+(+(x, y), z)] 205.19/60.05 205.19/60.05 [+(minus(+(x, 1())), 1())] = [1] x + [1] 205.19/60.05 >= [1] x + [1] 205.19/60.05 = [minus(x)] 205.19/60.05 205.19/60.05 [+(minus(1()), 1())] = [1] 205.19/60.05 >= [1] 205.19/60.05 = [0()] 205.19/60.05 205.19/60.05 [+(0(), y)] = [1] y + [1] 205.19/60.05 > [1] y + [0] 205.19/60.05 = [y] 205.19/60.05 205.19/60.05 205.19/60.05 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 205.19/60.05 205.19/60.05 We are left with following problem, upon which TcT provides the 205.19/60.05 certificate MAYBE. 205.19/60.05 205.19/60.05 Strict Trs: 205.19/60.05 { +(x, minus(y)) -> minus(+(minus(x), y)) 205.19/60.05 , +(x, +(y, z)) -> +(+(x, y), z) } 205.19/60.05 Weak Trs: 205.19/60.05 { minus(minus(x)) -> x 205.19/60.05 , minus(0()) -> 0() 205.19/60.05 , +(x, 0()) -> x 205.19/60.05 , +(minus(+(x, 1())), 1()) -> minus(x) 205.19/60.05 , +(minus(1()), 1()) -> 0() 205.19/60.05 , +(0(), y) -> y } 205.19/60.05 Obligation: 205.19/60.05 derivational complexity 205.19/60.05 Answer: 205.19/60.05 MAYBE 205.19/60.05 205.19/60.05 We use the processor 'matrix interpretation of dimension 2' to 205.19/60.05 orient following rules strictly. 205.19/60.05 205.19/60.05 Trs: { +(x, +(y, z)) -> +(+(x, y), z) } 205.19/60.05 205.19/60.05 The induced complexity on above rules (modulo remaining rules) is 205.19/60.05 YES(?,O(n^2)) . These rules are moved into the corresponding weak 205.19/60.05 component(s). 205.19/60.05 205.19/60.05 Sub-proof: 205.19/60.05 ---------- 205.19/60.05 TcT has computed the following triangular matrix interpretation. 205.19/60.05 205.19/60.05 [minus](x1) = [1 0] x1 + [0] 205.19/60.05 [0 1] [0] 205.19/60.05 205.19/60.05 [0] = [0] 205.19/60.05 [0] 205.19/60.05 205.19/60.05 [+](x1, x2) = [1 0] x1 + [1 1] x2 + [0] 205.19/60.05 [0 1] [0 1] [1] 205.19/60.05 205.19/60.05 [1] = [0] 205.19/60.05 [0] 205.19/60.05 205.19/60.05 The order satisfies the following ordering constraints: 205.19/60.05 205.19/60.05 [minus(minus(x))] = [1 0] x + [0] 205.19/60.05 [0 1] [0] 205.19/60.05 >= [1 0] x + [0] 205.19/60.05 [0 1] [0] 205.19/60.05 = [x] 205.19/60.05 205.19/60.05 [minus(0())] = [0] 205.19/60.05 [0] 205.19/60.05 >= [0] 205.19/60.05 [0] 205.19/60.05 = [0()] 205.19/60.05 205.19/60.05 [+(x, minus(y))] = [1 0] x + [1 1] y + [0] 205.19/60.05 [0 1] [0 1] [1] 205.19/60.05 >= [1 0] x + [1 1] y + [0] 205.19/60.05 [0 1] [0 1] [1] 205.19/60.05 = [minus(+(minus(x), y))] 205.19/60.05 205.19/60.05 [+(x, 0())] = [1 0] x + [0] 205.19/60.05 [0 1] [1] 205.19/60.05 >= [1 0] x + [0] 205.19/60.05 [0 1] [0] 205.19/60.05 = [x] 205.19/60.05 205.19/60.05 [+(x, +(y, z))] = [1 0] x + [1 1] y + [1 2] z + [1] 205.19/60.05 [0 1] [0 1] [0 1] [2] 205.19/60.05 > [1 0] x + [1 1] y + [1 1] z + [0] 205.19/60.05 [0 1] [0 1] [0 1] [2] 205.19/60.05 = [+(+(x, y), z)] 205.19/60.05 205.19/60.05 [+(minus(+(x, 1())), 1())] = [1 0] x + [0] 205.19/60.05 [0 1] [2] 205.19/60.05 >= [1 0] x + [0] 205.19/60.05 [0 1] [0] 205.19/60.05 = [minus(x)] 205.19/60.05 205.19/60.05 [+(minus(1()), 1())] = [0] 205.19/60.05 [1] 205.19/60.05 >= [0] 205.19/60.05 [0] 205.19/60.05 = [0()] 205.19/60.05 205.19/60.05 [+(0(), y)] = [1 1] y + [0] 205.19/60.05 [0 1] [1] 205.19/60.05 >= [1 0] y + [0] 205.19/60.05 [0 1] [0] 205.19/60.05 = [y] 205.19/60.05 205.19/60.05 205.19/60.05 We return to the main proof. 205.19/60.05 205.19/60.05 We are left with following problem, upon which TcT provides the 205.19/60.05 certificate MAYBE. 205.19/60.05 205.19/60.05 Strict Trs: { +(x, minus(y)) -> minus(+(minus(x), y)) } 205.19/60.05 Weak Trs: 205.19/60.05 { minus(minus(x)) -> x 205.19/60.05 , minus(0()) -> 0() 205.19/60.05 , +(x, 0()) -> x 205.19/60.05 , +(x, +(y, z)) -> +(+(x, y), z) 205.19/60.05 , +(minus(+(x, 1())), 1()) -> minus(x) 205.19/60.05 , +(minus(1()), 1()) -> 0() 205.19/60.05 , +(0(), y) -> y } 205.19/60.05 Obligation: 205.19/60.05 derivational complexity 205.19/60.05 Answer: 205.19/60.05 MAYBE 205.19/60.05 205.19/60.05 None of the processors succeeded. 205.19/60.05 205.19/60.05 Details of failed attempt(s): 205.19/60.05 ----------------------------- 205.19/60.05 1) 'empty' failed due to the following reason: 205.19/60.05 205.19/60.05 Empty strict component of the problem is NOT empty. 205.19/60.05 205.19/60.05 2) 'Fastest' failed due to the following reason: 205.19/60.05 205.19/60.05 None of the processors succeeded. 205.19/60.05 205.19/60.05 Details of failed attempt(s): 205.19/60.05 ----------------------------- 205.19/60.05 1) 'Fastest (timeout of 30 seconds)' failed due to the following 205.19/60.05 reason: 205.19/60.05 205.19/60.05 Computation stopped due to timeout after 30.0 seconds. 205.19/60.05 205.19/60.05 2) 'Fastest' failed due to the following reason: 205.19/60.05 205.19/60.05 None of the processors succeeded. 205.19/60.05 205.19/60.05 Details of failed attempt(s): 205.19/60.05 ----------------------------- 205.19/60.05 1) 'matrix interpretation of dimension 6' failed due to the 205.19/60.05 following reason: 205.19/60.05 205.19/60.05 The input cannot be shown compatible 205.19/60.05 205.19/60.05 2) 'matrix interpretation of dimension 5' failed due to the 205.19/60.05 following reason: 205.19/60.05 205.19/60.05 The input cannot be shown compatible 205.19/60.05 205.19/60.05 3) 'matrix interpretation of dimension 4' failed due to the 205.19/60.05 following reason: 205.19/60.05 205.19/60.05 The input cannot be shown compatible 205.19/60.05 205.19/60.05 4) 'matrix interpretation of dimension 3' failed due to the 205.19/60.05 following reason: 205.19/60.05 205.19/60.05 The input cannot be shown compatible 205.19/60.05 205.19/60.05 5) 'matrix interpretation of dimension 2' failed due to the 205.19/60.05 following reason: 205.19/60.05 205.19/60.05 The input cannot be shown compatible 205.19/60.05 205.19/60.05 6) 'matrix interpretation of dimension 1' failed due to the 205.19/60.05 following reason: 205.19/60.05 205.19/60.05 The input cannot be shown compatible 205.19/60.05 205.19/60.05 205.19/60.05 3) 'iteProgress' failed due to the following reason: 205.19/60.05 205.19/60.05 Fail 205.19/60.05 205.19/60.05 4) 'bsearch-matrix' failed due to the following reason: 205.19/60.05 205.19/60.05 The input cannot be shown compatible 205.19/60.05 205.19/60.05 205.19/60.05 205.19/60.05 3) 'iteProgress (timeout of 297 seconds)' failed due to the 205.19/60.05 following reason: 205.19/60.05 205.19/60.05 Fail 205.19/60.05 205.19/60.05 4) 'bsearch-matrix (timeout of 297 seconds)' failed due to the 205.19/60.05 following reason: 205.19/60.05 205.19/60.05 The input cannot be shown compatible 205.19/60.05 205.19/60.05 205.19/60.05 Arrrr.. 205.19/60.09 EOF