MAYBE 215.29/60.04 MAYBE 215.29/60.04 215.29/60.04 We are left with following problem, upon which TcT provides the 215.29/60.04 certificate MAYBE. 215.29/60.04 215.29/60.04 Strict Trs: 215.29/60.04 { a(f(), a(s(), x)) -> a(d(), a(f(), a(p(), a(s(), x)))) 215.29/60.04 , a(f(), 0()) -> a(s(), 0()) 215.29/60.04 , a(d(), a(s(), x)) -> a(s(), a(s(), a(d(), a(p(), a(s(), x))))) 215.29/60.04 , a(d(), 0()) -> 0() 215.29/60.04 , a(p(), a(s(), x)) -> x } 215.29/60.04 Obligation: 215.29/60.04 derivational complexity 215.29/60.04 Answer: 215.29/60.04 MAYBE 215.29/60.04 215.29/60.04 We uncurry the input using the following uncurry rules. 215.29/60.04 215.29/60.04 { a(f(), x_1) -> f_1(x_1) 215.29/60.04 , a(s(), x_1) -> s_1(x_1) 215.29/60.04 , a(d(), x_1) -> d_1(x_1) 215.29/60.04 , a(p(), x_1) -> p_1(x_1) } 215.29/60.04 215.29/60.04 We are left with following problem, upon which TcT provides the 215.29/60.04 certificate MAYBE. 215.29/60.04 215.29/60.04 Strict Trs: 215.29/60.04 { f_1(s_1(x)) -> d_1(f_1(p_1(s_1(x)))) 215.29/60.04 , f_1(0()) -> s_1(0()) 215.29/60.04 , d_1(s_1(x)) -> s_1(s_1(d_1(p_1(s_1(x))))) 215.29/60.04 , d_1(0()) -> 0() 215.29/60.04 , p_1(s_1(x)) -> x } 215.29/60.04 Weak Trs: 215.29/60.04 { a(f(), x_1) -> f_1(x_1) 215.29/60.04 , a(s(), x_1) -> s_1(x_1) 215.29/60.04 , a(d(), x_1) -> d_1(x_1) 215.29/60.04 , a(p(), x_1) -> p_1(x_1) } 215.29/60.04 Obligation: 215.29/60.04 derivational complexity 215.29/60.04 Answer: 215.29/60.04 MAYBE 215.29/60.04 215.29/60.04 None of the processors succeeded. 215.29/60.04 215.29/60.04 Details of failed attempt(s): 215.29/60.04 ----------------------------- 215.29/60.04 1) 'Fastest (timeout of 60 seconds)' failed due to the following 215.29/60.04 reason: 215.29/60.04 215.29/60.04 Computation stopped due to timeout after 60.0 seconds. 215.29/60.04 215.29/60.04 2) 'Inspecting Problem... (timeout of 297 seconds)' failed due to 215.29/60.04 the following reason: 215.29/60.04 215.29/60.04 We use the processor 'matrix interpretation of dimension 1' to 215.29/60.04 orient following rules strictly. 215.29/60.04 215.29/60.04 Trs: { f_1(0()) -> s_1(0()) } 215.29/60.04 215.29/60.04 The induced complexity on above rules (modulo remaining rules) is 215.29/60.04 YES(?,O(n^1)) . These rules are moved into the corresponding weak 215.29/60.04 component(s). 215.29/60.04 215.29/60.04 Sub-proof: 215.29/60.04 ---------- 215.29/60.04 TcT has computed the following triangular matrix interpretation. 215.29/60.04 215.29/60.04 [a](x1, x2) = [1] x1 + [1] x2 + [2] 215.29/60.04 215.29/60.04 [f_1](x1) = [1] x1 + [1] 215.29/60.04 215.29/60.04 [s_1](x1) = [1] x1 + [0] 215.29/60.04 215.29/60.04 [d_1](x1) = [1] x1 + [0] 215.29/60.04 215.29/60.04 [p_1](x1) = [1] x1 + [0] 215.29/60.04 215.29/60.04 [0] = [0] 215.29/60.04 215.29/60.04 [f] = [2] 215.29/60.04 215.29/60.04 [s] = [2] 215.29/60.04 215.29/60.04 [d] = [2] 215.29/60.04 215.29/60.04 [p] = [2] 215.29/60.04 215.29/60.04 The order satisfies the following ordering constraints: 215.29/60.04 215.29/60.04 [a(f(), x_1)] = [1] x_1 + [4] 215.29/60.04 > [1] x_1 + [1] 215.29/60.04 = [f_1(x_1)] 215.29/60.04 215.29/60.04 [a(s(), x_1)] = [1] x_1 + [4] 215.29/60.04 > [1] x_1 + [0] 215.29/60.04 = [s_1(x_1)] 215.29/60.04 215.29/60.04 [a(d(), x_1)] = [1] x_1 + [4] 215.29/60.04 > [1] x_1 + [0] 215.29/60.04 = [d_1(x_1)] 215.29/60.04 215.29/60.04 [a(p(), x_1)] = [1] x_1 + [4] 215.29/60.04 > [1] x_1 + [0] 215.29/60.04 = [p_1(x_1)] 215.29/60.04 215.29/60.04 [f_1(s_1(x))] = [1] x + [1] 215.29/60.04 >= [1] x + [1] 215.29/60.04 = [d_1(f_1(p_1(s_1(x))))] 215.29/60.04 215.29/60.04 [f_1(0())] = [1] 215.29/60.04 > [0] 215.29/60.04 = [s_1(0())] 215.29/60.04 215.29/60.04 [d_1(s_1(x))] = [1] x + [0] 215.29/60.04 >= [1] x + [0] 215.29/60.04 = [s_1(s_1(d_1(p_1(s_1(x)))))] 215.29/60.04 215.29/60.04 [d_1(0())] = [0] 215.29/60.04 >= [0] 215.29/60.04 = [0()] 215.29/60.04 215.29/60.04 [p_1(s_1(x))] = [1] x + [0] 215.29/60.04 >= [1] x + [0] 215.29/60.04 = [x] 215.29/60.04 215.29/60.04 215.29/60.04 We return to the main proof. 215.29/60.04 215.29/60.04 We are left with following problem, upon which TcT provides the 215.29/60.04 certificate MAYBE. 215.29/60.04 215.29/60.04 Strict Trs: 215.29/60.04 { f_1(s_1(x)) -> d_1(f_1(p_1(s_1(x)))) 215.29/60.04 , d_1(s_1(x)) -> s_1(s_1(d_1(p_1(s_1(x))))) 215.29/60.04 , d_1(0()) -> 0() 215.29/60.04 , p_1(s_1(x)) -> x } 215.29/60.04 Weak Trs: 215.29/60.04 { a(f(), x_1) -> f_1(x_1) 215.29/60.04 , a(s(), x_1) -> s_1(x_1) 215.29/60.04 , a(d(), x_1) -> d_1(x_1) 215.29/60.04 , a(p(), x_1) -> p_1(x_1) 215.29/60.04 , f_1(0()) -> s_1(0()) } 215.29/60.04 Obligation: 215.29/60.04 derivational complexity 215.29/60.04 Answer: 215.29/60.04 MAYBE 215.29/60.04 215.29/60.04 The weightgap principle applies (using the following nonconstant 215.29/60.04 growth matrix-interpretation) 215.29/60.04 215.29/60.04 TcT has computed the following triangular matrix interpretation. 215.29/60.04 Note that the diagonal of the component-wise maxima of 215.29/60.04 interpretation-entries contains no more than 1 non-zero entries. 215.29/60.04 215.29/60.04 [a](x1, x2) = [1] x1 + [1] x2 + [2] 215.29/60.04 215.29/60.04 [f_1](x1) = [1] x1 + [0] 215.29/60.04 215.29/60.04 [s_1](x1) = [1] x1 + [0] 215.29/60.04 215.29/60.04 [d_1](x1) = [1] x1 + [0] 215.29/60.04 215.29/60.04 [p_1](x1) = [1] x1 + [1] 215.29/60.04 215.29/60.04 [0] = [0] 215.29/60.04 215.29/60.04 [f] = [1] 215.29/60.04 215.29/60.04 [s] = [1] 215.29/60.04 215.29/60.04 [d] = [1] 215.29/60.04 215.29/60.04 [p] = [1] 215.29/60.04 215.29/60.04 The order satisfies the following ordering constraints: 215.29/60.04 215.29/60.04 [a(f(), x_1)] = [1] x_1 + [3] 215.29/60.04 > [1] x_1 + [0] 215.29/60.04 = [f_1(x_1)] 215.29/60.04 215.29/60.04 [a(s(), x_1)] = [1] x_1 + [3] 215.29/60.04 > [1] x_1 + [0] 215.29/60.04 = [s_1(x_1)] 215.29/60.04 215.29/60.04 [a(d(), x_1)] = [1] x_1 + [3] 215.29/60.04 > [1] x_1 + [0] 215.29/60.04 = [d_1(x_1)] 215.29/60.04 215.29/60.04 [a(p(), x_1)] = [1] x_1 + [3] 215.29/60.04 > [1] x_1 + [1] 215.29/60.04 = [p_1(x_1)] 215.29/60.04 215.29/60.04 [f_1(s_1(x))] = [1] x + [0] 215.29/60.04 ? [1] x + [1] 215.29/60.04 = [d_1(f_1(p_1(s_1(x))))] 215.29/60.04 215.29/60.04 [f_1(0())] = [0] 215.29/60.04 >= [0] 215.29/60.04 = [s_1(0())] 215.29/60.04 215.29/60.04 [d_1(s_1(x))] = [1] x + [0] 215.29/60.04 ? [1] x + [1] 215.29/60.04 = [s_1(s_1(d_1(p_1(s_1(x)))))] 215.29/60.04 215.29/60.04 [d_1(0())] = [0] 215.29/60.04 >= [0] 215.29/60.04 = [0()] 215.29/60.04 215.29/60.04 [p_1(s_1(x))] = [1] x + [1] 215.29/60.04 > [1] x + [0] 215.29/60.04 = [x] 215.29/60.04 215.29/60.04 215.29/60.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 215.29/60.04 215.29/60.04 We are left with following problem, upon which TcT provides the 215.29/60.04 certificate MAYBE. 215.29/60.04 215.29/60.04 Strict Trs: 215.29/60.04 { f_1(s_1(x)) -> d_1(f_1(p_1(s_1(x)))) 215.29/60.04 , d_1(s_1(x)) -> s_1(s_1(d_1(p_1(s_1(x))))) 215.29/60.04 , d_1(0()) -> 0() } 215.29/60.04 Weak Trs: 215.29/60.04 { a(f(), x_1) -> f_1(x_1) 215.29/60.04 , a(s(), x_1) -> s_1(x_1) 215.29/60.04 , a(d(), x_1) -> d_1(x_1) 215.29/60.04 , a(p(), x_1) -> p_1(x_1) 215.29/60.04 , f_1(0()) -> s_1(0()) 215.29/60.04 , p_1(s_1(x)) -> x } 215.29/60.04 Obligation: 215.29/60.04 derivational complexity 215.29/60.04 Answer: 215.29/60.04 MAYBE 215.29/60.04 215.29/60.04 The weightgap principle applies (using the following nonconstant 215.29/60.04 growth matrix-interpretation) 215.29/60.04 215.29/60.04 TcT has computed the following triangular matrix interpretation. 215.29/60.04 Note that the diagonal of the component-wise maxima of 215.29/60.04 interpretation-entries contains no more than 1 non-zero entries. 215.29/60.04 215.29/60.04 [a](x1, x2) = [1] x1 + [1] x2 + [2] 215.29/60.04 215.29/60.04 [f_1](x1) = [1] x1 + [0] 215.29/60.04 215.29/60.04 [s_1](x1) = [1] x1 + [0] 215.29/60.04 215.29/60.04 [d_1](x1) = [1] x1 + [1] 215.29/60.04 215.29/60.04 [p_1](x1) = [1] x1 + [0] 215.29/60.04 215.29/60.04 [0] = [0] 215.29/60.04 215.29/60.04 [f] = [1] 215.29/60.04 215.29/60.04 [s] = [1] 215.29/60.04 215.29/60.04 [d] = [1] 215.29/60.04 215.29/60.04 [p] = [1] 215.29/60.04 215.29/60.04 The order satisfies the following ordering constraints: 215.29/60.04 215.29/60.04 [a(f(), x_1)] = [1] x_1 + [3] 215.29/60.04 > [1] x_1 + [0] 215.29/60.04 = [f_1(x_1)] 215.29/60.04 215.29/60.04 [a(s(), x_1)] = [1] x_1 + [3] 215.29/60.04 > [1] x_1 + [0] 215.29/60.04 = [s_1(x_1)] 215.29/60.04 215.29/60.04 [a(d(), x_1)] = [1] x_1 + [3] 215.29/60.04 > [1] x_1 + [1] 215.29/60.04 = [d_1(x_1)] 215.29/60.04 215.29/60.04 [a(p(), x_1)] = [1] x_1 + [3] 215.29/60.04 > [1] x_1 + [0] 215.29/60.04 = [p_1(x_1)] 215.29/60.04 215.29/60.04 [f_1(s_1(x))] = [1] x + [0] 215.29/60.04 ? [1] x + [1] 215.29/60.04 = [d_1(f_1(p_1(s_1(x))))] 215.29/60.04 215.29/60.04 [f_1(0())] = [0] 215.29/60.04 >= [0] 215.29/60.04 = [s_1(0())] 215.29/60.04 215.29/60.04 [d_1(s_1(x))] = [1] x + [1] 215.29/60.04 >= [1] x + [1] 215.29/60.04 = [s_1(s_1(d_1(p_1(s_1(x)))))] 215.29/60.04 215.29/60.04 [d_1(0())] = [1] 215.29/60.04 > [0] 215.29/60.04 = [0()] 215.29/60.04 215.29/60.04 [p_1(s_1(x))] = [1] x + [0] 215.29/60.04 >= [1] x + [0] 215.29/60.04 = [x] 215.29/60.04 215.29/60.04 215.29/60.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 215.29/60.04 215.29/60.04 We are left with following problem, upon which TcT provides the 215.29/60.04 certificate MAYBE. 215.29/60.04 215.29/60.04 Strict Trs: 215.29/60.04 { f_1(s_1(x)) -> d_1(f_1(p_1(s_1(x)))) 215.29/60.04 , d_1(s_1(x)) -> s_1(s_1(d_1(p_1(s_1(x))))) } 215.29/60.04 Weak Trs: 215.29/60.04 { a(f(), x_1) -> f_1(x_1) 215.29/60.04 , a(s(), x_1) -> s_1(x_1) 215.29/60.04 , a(d(), x_1) -> d_1(x_1) 215.29/60.04 , a(p(), x_1) -> p_1(x_1) 215.29/60.04 , f_1(0()) -> s_1(0()) 215.29/60.04 , d_1(0()) -> 0() 215.29/60.04 , p_1(s_1(x)) -> x } 215.29/60.04 Obligation: 215.29/60.04 derivational complexity 215.29/60.04 Answer: 215.29/60.04 MAYBE 215.29/60.04 215.29/60.04 None of the processors succeeded. 215.29/60.04 215.29/60.04 Details of failed attempt(s): 215.29/60.04 ----------------------------- 215.29/60.04 1) 'empty' failed due to the following reason: 215.29/60.04 215.29/60.04 Empty strict component of the problem is NOT empty. 215.29/60.04 215.29/60.04 2) 'Fastest' failed due to the following reason: 215.29/60.04 215.29/60.04 None of the processors succeeded. 215.29/60.04 215.29/60.04 Details of failed attempt(s): 215.29/60.04 ----------------------------- 215.29/60.04 1) 'bsearch-matrix' failed due to the following reason: 215.29/60.04 215.29/60.04 The input cannot be shown compatible 215.29/60.04 215.29/60.04 2) 'iteProgress' failed due to the following reason: 215.29/60.04 215.29/60.04 Fail 215.29/60.04 215.29/60.04 3) 'Fastest' failed due to the following reason: 215.29/60.04 215.29/60.04 None of the processors succeeded. 215.29/60.04 215.29/60.04 Details of failed attempt(s): 215.29/60.04 ----------------------------- 215.29/60.04 1) 'matrix interpretation of dimension 6' failed due to the 215.29/60.04 following reason: 215.29/60.04 215.29/60.04 The input cannot be shown compatible 215.29/60.04 215.29/60.04 2) 'matrix interpretation of dimension 5' failed due to the 215.29/60.04 following reason: 215.29/60.04 215.29/60.04 The input cannot be shown compatible 215.29/60.04 215.29/60.04 3) 'matrix interpretation of dimension 4' failed due to the 215.29/60.04 following reason: 215.29/60.04 215.29/60.04 The input cannot be shown compatible 215.29/60.04 215.29/60.04 4) 'matrix interpretation of dimension 3' failed due to the 215.29/60.04 following reason: 215.29/60.04 215.29/60.04 The input cannot be shown compatible 215.29/60.04 215.29/60.04 5) 'matrix interpretation of dimension 2' failed due to the 215.29/60.04 following reason: 215.29/60.04 215.29/60.04 The input cannot be shown compatible 215.29/60.04 215.29/60.04 6) 'matrix interpretation of dimension 1' failed due to the 215.29/60.04 following reason: 215.29/60.04 215.29/60.04 The input cannot be shown compatible 215.29/60.04 215.29/60.04 215.29/60.04 4) 'Fastest (timeout of 30 seconds)' failed due to the following 215.29/60.04 reason: 215.29/60.04 215.29/60.04 Computation stopped due to timeout after 30.0 seconds. 215.29/60.04 215.29/60.04 215.29/60.04 215.29/60.04 3) 'iteProgress (timeout of 297 seconds)' failed due to the 215.29/60.04 following reason: 215.29/60.04 215.29/60.04 Fail 215.29/60.04 215.29/60.04 4) 'bsearch-matrix (timeout of 297 seconds)' failed due to the 215.29/60.04 following reason: 215.29/60.04 215.29/60.04 The input cannot be shown compatible 215.29/60.04 215.29/60.04 215.29/60.04 Arrrr.. 215.29/60.05 EOF