MAYBE 207.67/60.03 MAYBE 207.67/60.03 207.67/60.03 We are left with following problem, upon which TcT provides the 207.67/60.03 certificate MAYBE. 207.67/60.03 207.67/60.03 Strict Trs: 207.67/60.03 { app(nil(), y) -> y 207.67/60.03 , app(add(n, x), y) -> add(n, app(x, y)) 207.67/60.03 , reverse(nil()) -> nil() 207.67/60.03 , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) 207.67/60.03 , shuffle(nil()) -> nil() 207.67/60.03 , shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) } 207.67/60.03 Obligation: 207.67/60.03 derivational complexity 207.67/60.03 Answer: 207.67/60.03 MAYBE 207.67/60.03 207.67/60.03 None of the processors succeeded. 207.67/60.03 207.67/60.03 Details of failed attempt(s): 207.67/60.03 ----------------------------- 207.67/60.03 1) 'Fastest (timeout of 60 seconds)' failed due to the following 207.67/60.03 reason: 207.67/60.03 207.67/60.03 Computation stopped due to timeout after 60.0 seconds. 207.67/60.03 207.67/60.03 2) 'Inspecting Problem... (timeout of 297 seconds)' failed due to 207.67/60.03 the following reason: 207.67/60.03 207.67/60.03 We use the processor 'matrix interpretation of dimension 1' to 207.67/60.03 orient following rules strictly. 207.67/60.03 207.67/60.03 Trs: { shuffle(nil()) -> nil() } 207.67/60.03 207.67/60.03 The induced complexity on above rules (modulo remaining rules) is 207.67/60.03 YES(?,O(n^1)) . These rules are moved into the corresponding weak 207.67/60.03 component(s). 207.67/60.03 207.67/60.03 Sub-proof: 207.67/60.03 ---------- 207.67/60.03 TcT has computed the following triangular matrix interpretation. 207.67/60.03 207.67/60.03 [app](x1, x2) = [1] x1 + [1] x2 + [0] 207.67/60.03 207.67/60.03 [nil] = [0] 207.67/60.03 207.67/60.03 [add](x1, x2) = [1] x1 + [1] x2 + [0] 207.67/60.03 207.67/60.03 [reverse](x1) = [1] x1 + [0] 207.67/60.03 207.67/60.03 [shuffle](x1) = [1] x1 + [1] 207.67/60.03 207.67/60.03 The order satisfies the following ordering constraints: 207.67/60.03 207.67/60.03 [app(nil(), y)] = [1] y + [0] 207.67/60.03 >= [1] y + [0] 207.67/60.03 = [y] 207.67/60.03 207.67/60.03 [app(add(n, x), y)] = [1] y + [1] n + [1] x + [0] 207.67/60.03 >= [1] y + [1] n + [1] x + [0] 207.67/60.03 = [add(n, app(x, y))] 207.67/60.03 207.67/60.03 [reverse(nil())] = [0] 207.67/60.03 >= [0] 207.67/60.03 = [nil()] 207.67/60.03 207.67/60.03 [reverse(add(n, x))] = [1] n + [1] x + [0] 207.67/60.03 >= [1] n + [1] x + [0] 207.67/60.03 = [app(reverse(x), add(n, nil()))] 207.67/60.03 207.67/60.03 [shuffle(nil())] = [1] 207.67/60.03 > [0] 207.67/60.03 = [nil()] 207.67/60.03 207.67/60.03 [shuffle(add(n, x))] = [1] n + [1] x + [1] 207.67/60.03 >= [1] n + [1] x + [1] 207.67/60.03 = [add(n, shuffle(reverse(x)))] 207.67/60.03 207.67/60.03 207.67/60.03 We return to the main proof. 207.67/60.03 207.67/60.03 We are left with following problem, upon which TcT provides the 207.67/60.03 certificate MAYBE. 207.67/60.03 207.67/60.03 Strict Trs: 207.67/60.03 { app(nil(), y) -> y 207.67/60.03 , app(add(n, x), y) -> add(n, app(x, y)) 207.67/60.03 , reverse(nil()) -> nil() 207.67/60.03 , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) 207.67/60.03 , shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) } 207.67/60.03 Weak Trs: { shuffle(nil()) -> nil() } 207.67/60.03 Obligation: 207.67/60.03 derivational complexity 207.67/60.03 Answer: 207.67/60.03 MAYBE 207.67/60.03 207.67/60.03 The weightgap principle applies (using the following nonconstant 207.67/60.03 growth matrix-interpretation) 207.67/60.03 207.67/60.03 TcT has computed the following triangular matrix interpretation. 207.67/60.03 Note that the diagonal of the component-wise maxima of 207.67/60.03 interpretation-entries contains no more than 1 non-zero entries. 207.67/60.03 207.67/60.03 [app](x1, x2) = [1] x1 + [1] x2 + [0] 207.67/60.03 207.67/60.03 [nil] = [0] 207.67/60.03 207.67/60.03 [add](x1, x2) = [1] x1 + [1] x2 + [0] 207.67/60.03 207.67/60.03 [reverse](x1) = [1] x1 + [1] 207.67/60.03 207.67/60.03 [shuffle](x1) = [1] x1 + [0] 207.67/60.03 207.67/60.03 The order satisfies the following ordering constraints: 207.67/60.03 207.67/60.03 [app(nil(), y)] = [1] y + [0] 207.67/60.03 >= [1] y + [0] 207.67/60.03 = [y] 207.67/60.03 207.67/60.03 [app(add(n, x), y)] = [1] y + [1] n + [1] x + [0] 207.67/60.03 >= [1] y + [1] n + [1] x + [0] 207.67/60.03 = [add(n, app(x, y))] 207.67/60.03 207.67/60.03 [reverse(nil())] = [1] 207.67/60.03 > [0] 207.67/60.03 = [nil()] 207.67/60.03 207.67/60.03 [reverse(add(n, x))] = [1] n + [1] x + [1] 207.67/60.03 >= [1] n + [1] x + [1] 207.67/60.03 = [app(reverse(x), add(n, nil()))] 207.67/60.03 207.67/60.03 [shuffle(nil())] = [0] 207.67/60.03 >= [0] 207.67/60.03 = [nil()] 207.67/60.03 207.67/60.03 [shuffle(add(n, x))] = [1] n + [1] x + [0] 207.67/60.03 ? [1] n + [1] x + [1] 207.67/60.03 = [add(n, shuffle(reverse(x)))] 207.67/60.03 207.67/60.03 207.67/60.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 207.67/60.03 207.67/60.03 We are left with following problem, upon which TcT provides the 207.67/60.03 certificate MAYBE. 207.67/60.03 207.67/60.03 Strict Trs: 207.67/60.03 { app(nil(), y) -> y 207.67/60.03 , app(add(n, x), y) -> add(n, app(x, y)) 207.67/60.03 , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) 207.67/60.03 , shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) } 207.67/60.03 Weak Trs: 207.67/60.03 { reverse(nil()) -> nil() 207.67/60.03 , shuffle(nil()) -> nil() } 207.67/60.03 Obligation: 207.67/60.03 derivational complexity 207.67/60.03 Answer: 207.67/60.03 MAYBE 207.67/60.03 207.67/60.03 The weightgap principle applies (using the following nonconstant 207.67/60.03 growth matrix-interpretation) 207.67/60.03 207.67/60.03 TcT has computed the following triangular matrix interpretation. 207.67/60.03 Note that the diagonal of the component-wise maxima of 207.67/60.03 interpretation-entries contains no more than 1 non-zero entries. 207.67/60.03 207.67/60.03 [app](x1, x2) = [1] x1 + [1] x2 + [1] 207.67/60.03 207.67/60.03 [nil] = [0] 207.67/60.03 207.67/60.03 [add](x1, x2) = [1] x1 + [1] x2 + [0] 207.67/60.03 207.67/60.03 [reverse](x1) = [1] x1 + [0] 207.67/60.03 207.67/60.03 [shuffle](x1) = [1] x1 + [0] 207.67/60.03 207.67/60.03 The order satisfies the following ordering constraints: 207.67/60.03 207.67/60.03 [app(nil(), y)] = [1] y + [1] 207.67/60.03 > [1] y + [0] 207.67/60.03 = [y] 207.67/60.03 207.67/60.03 [app(add(n, x), y)] = [1] y + [1] n + [1] x + [1] 207.67/60.03 >= [1] y + [1] n + [1] x + [1] 207.67/60.03 = [add(n, app(x, y))] 207.67/60.03 207.67/60.03 [reverse(nil())] = [0] 207.67/60.03 >= [0] 207.67/60.03 = [nil()] 207.67/60.03 207.67/60.03 [reverse(add(n, x))] = [1] n + [1] x + [0] 207.67/60.03 ? [1] n + [1] x + [1] 207.67/60.03 = [app(reverse(x), add(n, nil()))] 207.67/60.03 207.67/60.03 [shuffle(nil())] = [0] 207.67/60.03 >= [0] 207.67/60.03 = [nil()] 207.67/60.03 207.67/60.03 [shuffle(add(n, x))] = [1] n + [1] x + [0] 207.67/60.03 >= [1] n + [1] x + [0] 207.67/60.03 = [add(n, shuffle(reverse(x)))] 207.67/60.03 207.67/60.03 207.67/60.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 207.67/60.03 207.67/60.03 We are left with following problem, upon which TcT provides the 207.67/60.03 certificate MAYBE. 207.67/60.03 207.67/60.03 Strict Trs: 207.67/60.03 { app(add(n, x), y) -> add(n, app(x, y)) 207.67/60.03 , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) 207.67/60.03 , shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) } 207.67/60.03 Weak Trs: 207.67/60.03 { app(nil(), y) -> y 207.67/60.03 , reverse(nil()) -> nil() 207.67/60.03 , shuffle(nil()) -> nil() } 207.67/60.03 Obligation: 207.67/60.03 derivational complexity 207.67/60.03 Answer: 207.67/60.03 MAYBE 207.67/60.03 207.67/60.03 We use the processor 'matrix interpretation of dimension 2' to 207.67/60.03 orient following rules strictly. 207.67/60.03 207.67/60.03 Trs: { shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) } 207.67/60.03 207.67/60.03 The induced complexity on above rules (modulo remaining rules) is 207.67/60.03 YES(?,O(n^2)) . These rules are moved into the corresponding weak 207.67/60.03 component(s). 207.67/60.03 207.67/60.03 Sub-proof: 207.67/60.03 ---------- 207.67/60.03 TcT has computed the following triangular matrix interpretation. 207.67/60.03 207.67/60.03 [app](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 207.67/60.03 [0 1] [0 1] [0] 207.67/60.03 207.67/60.03 [nil] = [0] 207.67/60.03 [0] 207.67/60.03 207.67/60.03 [add](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 207.67/60.03 [0 0] [0 1] [1] 207.67/60.03 207.67/60.03 [reverse](x1) = [1 0] x1 + [0] 207.67/60.03 [0 1] [0] 207.67/60.03 207.67/60.03 [shuffle](x1) = [1 2] x1 + [0] 207.67/60.03 [0 1] [0] 207.67/60.03 207.67/60.03 The order satisfies the following ordering constraints: 207.67/60.03 207.67/60.03 [app(nil(), y)] = [1 0] y + [0] 207.67/60.03 [0 1] [0] 207.67/60.03 >= [1 0] y + [0] 207.67/60.03 [0 1] [0] 207.67/60.03 = [y] 207.67/60.03 207.67/60.03 [app(add(n, x), y)] = [1 0] y + [1 0] n + [1 0] x + [0] 207.67/60.03 [0 1] [0 0] [0 1] [1] 207.67/60.03 >= [1 0] y + [1 0] n + [1 0] x + [0] 207.67/60.03 [0 1] [0 0] [0 1] [1] 207.67/60.03 = [add(n, app(x, y))] 207.67/60.03 207.67/60.03 [reverse(nil())] = [0] 207.67/60.03 [0] 207.67/60.03 >= [0] 207.67/60.03 [0] 207.67/60.03 = [nil()] 207.67/60.03 207.67/60.03 [reverse(add(n, x))] = [1 0] n + [1 0] x + [0] 207.67/60.03 [0 0] [0 1] [1] 207.67/60.03 >= [1 0] n + [1 0] x + [0] 207.67/60.03 [0 0] [0 1] [1] 207.67/60.03 = [app(reverse(x), add(n, nil()))] 207.67/60.03 207.67/60.03 [shuffle(nil())] = [0] 207.67/60.03 [0] 207.67/60.03 >= [0] 207.67/60.03 [0] 207.67/60.03 = [nil()] 207.67/60.03 207.67/60.03 [shuffle(add(n, x))] = [1 0] n + [1 2] x + [2] 207.67/60.03 [0 0] [0 1] [1] 207.67/60.03 > [1 0] n + [1 2] x + [0] 207.67/60.03 [0 0] [0 1] [1] 207.67/60.03 = [add(n, shuffle(reverse(x)))] 207.67/60.03 207.67/60.03 207.67/60.03 We return to the main proof. 207.67/60.03 207.67/60.03 We are left with following problem, upon which TcT provides the 207.67/60.03 certificate MAYBE. 207.67/60.03 207.67/60.03 Strict Trs: 207.67/60.03 { app(add(n, x), y) -> add(n, app(x, y)) 207.67/60.03 , reverse(add(n, x)) -> app(reverse(x), add(n, nil())) } 207.67/60.03 Weak Trs: 207.67/60.03 { app(nil(), y) -> y 207.67/60.03 , reverse(nil()) -> nil() 207.67/60.03 , shuffle(nil()) -> nil() 207.67/60.03 , shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) } 207.67/60.03 Obligation: 207.67/60.03 derivational complexity 207.67/60.03 Answer: 207.67/60.03 MAYBE 207.67/60.03 207.67/60.03 None of the processors succeeded. 207.67/60.03 207.67/60.03 Details of failed attempt(s): 207.67/60.03 ----------------------------- 207.67/60.03 1) 'empty' failed due to the following reason: 207.67/60.03 207.67/60.03 Empty strict component of the problem is NOT empty. 207.67/60.03 207.67/60.03 2) 'Fastest' failed due to the following reason: 207.67/60.03 207.67/60.03 None of the processors succeeded. 207.67/60.03 207.67/60.03 Details of failed attempt(s): 207.67/60.03 ----------------------------- 207.67/60.03 1) 'Fastest (timeout of 30 seconds)' failed due to the following 207.67/60.03 reason: 207.67/60.03 207.67/60.03 Computation stopped due to timeout after 30.0 seconds. 207.67/60.03 207.67/60.03 2) 'Fastest' failed due to the following reason: 207.67/60.03 207.67/60.03 None of the processors succeeded. 207.67/60.03 207.67/60.03 Details of failed attempt(s): 207.67/60.03 ----------------------------- 207.67/60.03 1) 'matrix interpretation of dimension 6' failed due to the 207.67/60.03 following reason: 207.67/60.03 207.67/60.03 The input cannot be shown compatible 207.67/60.03 207.67/60.03 2) 'matrix interpretation of dimension 5' failed due to the 207.67/60.03 following reason: 207.67/60.03 207.67/60.03 The input cannot be shown compatible 207.67/60.03 207.67/60.03 3) 'matrix interpretation of dimension 4' failed due to the 207.67/60.03 following reason: 207.67/60.03 207.67/60.03 The input cannot be shown compatible 207.67/60.03 207.67/60.03 4) 'matrix interpretation of dimension 3' failed due to the 207.67/60.03 following reason: 207.67/60.03 207.67/60.03 The input cannot be shown compatible 207.67/60.03 207.67/60.03 5) 'matrix interpretation of dimension 2' failed due to the 207.67/60.03 following reason: 207.67/60.03 207.67/60.03 The input cannot be shown compatible 207.67/60.03 207.67/60.03 6) 'matrix interpretation of dimension 1' failed due to the 207.67/60.03 following reason: 207.67/60.03 207.67/60.03 The input cannot be shown compatible 207.67/60.03 207.67/60.03 207.67/60.03 3) 'iteProgress' failed due to the following reason: 207.67/60.03 207.67/60.03 Fail 207.67/60.03 207.67/60.03 4) 'bsearch-matrix' failed due to the following reason: 207.67/60.03 207.67/60.03 The input cannot be shown compatible 207.67/60.03 207.67/60.03 207.67/60.03 207.67/60.03 3) 'iteProgress (timeout of 297 seconds)' failed due to the 207.67/60.03 following reason: 207.67/60.03 207.67/60.03 Fail 207.67/60.03 207.67/60.03 4) 'bsearch-matrix (timeout of 297 seconds)' failed due to the 207.67/60.03 following reason: 207.67/60.03 207.67/60.03 The input cannot be shown compatible 207.67/60.03 207.67/60.03 207.67/60.03 Arrrr.. 207.67/60.04 EOF