MAYBE 202.07/60.03 MAYBE 202.07/60.03 202.07/60.03 We are left with following problem, upon which TcT provides the 202.07/60.03 certificate MAYBE. 202.07/60.03 202.07/60.03 Strict Trs: 202.07/60.03 { a__f(X) -> a__if(mark(X), c(), f(true())) 202.07/60.03 , a__f(X) -> f(X) 202.07/60.03 , a__if(X1, X2, X3) -> if(X1, X2, X3) 202.07/60.03 , a__if(true(), X, Y) -> mark(X) 202.07/60.03 , a__if(false(), X, Y) -> mark(Y) 202.07/60.03 , mark(c()) -> c() 202.07/60.03 , mark(f(X)) -> a__f(mark(X)) 202.07/60.03 , mark(true()) -> true() 202.07/60.03 , mark(false()) -> false() 202.07/60.03 , mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3) } 202.07/60.03 Obligation: 202.07/60.03 derivational complexity 202.07/60.03 Answer: 202.07/60.03 MAYBE 202.07/60.03 202.07/60.03 None of the processors succeeded. 202.07/60.03 202.07/60.03 Details of failed attempt(s): 202.07/60.03 ----------------------------- 202.07/60.03 1) 'Fastest (timeout of 60 seconds)' failed due to the following 202.07/60.03 reason: 202.07/60.03 202.07/60.03 Computation stopped due to timeout after 60.0 seconds. 202.07/60.03 202.07/60.03 2) 'Inspecting Problem... (timeout of 297 seconds)' failed due to 202.07/60.03 the following reason: 202.07/60.03 202.07/60.03 The weightgap principle applies (using the following nonconstant 202.07/60.03 growth matrix-interpretation) 202.07/60.03 202.07/60.03 TcT has computed the following triangular matrix interpretation. 202.07/60.03 Note that the diagonal of the component-wise maxima of 202.07/60.03 interpretation-entries contains no more than 1 non-zero entries. 202.07/60.03 202.07/60.03 [a__f](x1) = [1] x1 + [1] 202.07/60.03 202.07/60.03 [a__if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] 202.07/60.03 202.07/60.03 [mark](x1) = [1] x1 + [0] 202.07/60.03 202.07/60.03 [c] = [0] 202.07/60.03 202.07/60.03 [f](x1) = [1] x1 + [0] 202.07/60.03 202.07/60.03 [true] = [0] 202.07/60.03 202.07/60.03 [false] = [0] 202.07/60.03 202.07/60.03 [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1] 202.07/60.03 202.07/60.03 The order satisfies the following ordering constraints: 202.07/60.03 202.07/60.03 [a__f(X)] = [1] X + [1] 202.07/60.03 > [1] X + [0] 202.07/60.03 = [a__if(mark(X), c(), f(true()))] 202.07/60.03 202.07/60.03 [a__f(X)] = [1] X + [1] 202.07/60.03 > [1] X + [0] 202.07/60.03 = [f(X)] 202.07/60.03 202.07/60.03 [a__if(X1, X2, X3)] = [1] X1 + [1] X2 + [1] X3 + [0] 202.07/60.03 ? [1] X1 + [1] X2 + [1] X3 + [1] 202.07/60.03 = [if(X1, X2, X3)] 202.07/60.03 202.07/60.03 [a__if(true(), X, Y)] = [1] X + [1] Y + [0] 202.07/60.03 >= [1] X + [0] 202.07/60.03 = [mark(X)] 202.07/60.03 202.07/60.03 [a__if(false(), X, Y)] = [1] X + [1] Y + [0] 202.07/60.03 >= [1] Y + [0] 202.07/60.03 = [mark(Y)] 202.07/60.03 202.07/60.03 [mark(c())] = [0] 202.07/60.03 >= [0] 202.07/60.03 = [c()] 202.07/60.03 202.07/60.03 [mark(f(X))] = [1] X + [0] 202.07/60.03 ? [1] X + [1] 202.07/60.03 = [a__f(mark(X))] 202.07/60.03 202.07/60.03 [mark(true())] = [0] 202.07/60.03 >= [0] 202.07/60.03 = [true()] 202.07/60.03 202.07/60.03 [mark(false())] = [0] 202.07/60.03 >= [0] 202.07/60.03 = [false()] 202.07/60.03 202.07/60.03 [mark(if(X1, X2, X3))] = [1] X1 + [1] X2 + [1] X3 + [1] 202.07/60.03 > [1] X1 + [1] X2 + [1] X3 + [0] 202.07/60.03 = [a__if(mark(X1), mark(X2), X3)] 202.07/60.03 202.07/60.03 202.07/60.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 202.07/60.03 202.07/60.03 We are left with following problem, upon which TcT provides the 202.07/60.03 certificate MAYBE. 202.07/60.03 202.07/60.03 Strict Trs: 202.07/60.03 { a__if(X1, X2, X3) -> if(X1, X2, X3) 202.07/60.03 , a__if(true(), X, Y) -> mark(X) 202.07/60.03 , a__if(false(), X, Y) -> mark(Y) 202.07/60.03 , mark(c()) -> c() 202.07/60.03 , mark(f(X)) -> a__f(mark(X)) 202.07/60.03 , mark(true()) -> true() 202.07/60.03 , mark(false()) -> false() } 202.07/60.03 Weak Trs: 202.07/60.03 { a__f(X) -> a__if(mark(X), c(), f(true())) 202.07/60.03 , a__f(X) -> f(X) 202.07/60.03 , mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3) } 202.07/60.03 Obligation: 202.07/60.03 derivational complexity 202.07/60.03 Answer: 202.07/60.03 MAYBE 202.07/60.03 202.07/60.03 We use the processor 'matrix interpretation of dimension 1' to 202.07/60.03 orient following rules strictly. 202.07/60.03 202.07/60.03 Trs: { a__if(false(), X, Y) -> mark(Y) } 202.07/60.03 202.07/60.03 The induced complexity on above rules (modulo remaining rules) is 202.07/60.03 YES(?,O(n^1)) . These rules are moved into the corresponding weak 202.07/60.03 component(s). 202.07/60.03 202.07/60.03 Sub-proof: 202.07/60.03 ---------- 202.07/60.03 TcT has computed the following triangular matrix interpretation. 202.07/60.03 202.07/60.03 [a__f](x1) = [1] x1 + [0] 202.07/60.03 202.07/60.03 [a__if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] 202.07/60.03 202.07/60.03 [mark](x1) = [1] x1 + [0] 202.07/60.03 202.07/60.03 [c] = [0] 202.07/60.03 202.07/60.03 [f](x1) = [1] x1 + [0] 202.07/60.03 202.07/60.03 [true] = [0] 202.07/60.03 202.07/60.03 [false] = [1] 202.07/60.03 202.07/60.03 [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] 202.07/60.03 202.07/60.03 The order satisfies the following ordering constraints: 202.07/60.03 202.07/60.03 [a__f(X)] = [1] X + [0] 202.07/60.03 >= [1] X + [0] 202.07/60.03 = [a__if(mark(X), c(), f(true()))] 202.07/60.03 202.07/60.03 [a__f(X)] = [1] X + [0] 202.07/60.03 >= [1] X + [0] 202.07/60.03 = [f(X)] 202.07/60.03 202.07/60.03 [a__if(X1, X2, X3)] = [1] X1 + [1] X2 + [1] X3 + [0] 202.07/60.03 >= [1] X1 + [1] X2 + [1] X3 + [0] 202.07/60.03 = [if(X1, X2, X3)] 202.07/60.03 202.07/60.03 [a__if(true(), X, Y)] = [1] X + [1] Y + [0] 202.07/60.03 >= [1] X + [0] 202.07/60.03 = [mark(X)] 202.07/60.03 202.07/60.03 [a__if(false(), X, Y)] = [1] X + [1] Y + [1] 202.07/60.03 > [1] Y + [0] 202.07/60.03 = [mark(Y)] 202.07/60.03 202.07/60.03 [mark(c())] = [0] 202.07/60.03 >= [0] 202.07/60.03 = [c()] 202.07/60.03 202.07/60.03 [mark(f(X))] = [1] X + [0] 202.07/60.03 >= [1] X + [0] 202.07/60.03 = [a__f(mark(X))] 202.07/60.03 202.07/60.03 [mark(true())] = [0] 202.07/60.03 >= [0] 202.07/60.03 = [true()] 202.07/60.03 202.07/60.03 [mark(false())] = [1] 202.07/60.03 >= [1] 202.07/60.03 = [false()] 202.07/60.03 202.07/60.03 [mark(if(X1, X2, X3))] = [1] X1 + [1] X2 + [1] X3 + [0] 202.07/60.03 >= [1] X1 + [1] X2 + [1] X3 + [0] 202.07/60.03 = [a__if(mark(X1), mark(X2), X3)] 202.07/60.03 202.07/60.03 202.07/60.03 We return to the main proof. 202.07/60.03 202.07/60.03 We are left with following problem, upon which TcT provides the 202.07/60.03 certificate MAYBE. 202.07/60.03 202.07/60.03 Strict Trs: 202.07/60.03 { a__if(X1, X2, X3) -> if(X1, X2, X3) 202.07/60.03 , a__if(true(), X, Y) -> mark(X) 202.07/60.03 , mark(c()) -> c() 202.07/60.03 , mark(f(X)) -> a__f(mark(X)) 202.07/60.03 , mark(true()) -> true() 202.07/60.03 , mark(false()) -> false() } 202.07/60.03 Weak Trs: 202.07/60.03 { a__f(X) -> a__if(mark(X), c(), f(true())) 202.07/60.03 , a__f(X) -> f(X) 202.07/60.03 , a__if(false(), X, Y) -> mark(Y) 202.07/60.03 , mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3) } 202.07/60.03 Obligation: 202.07/60.03 derivational complexity 202.07/60.03 Answer: 202.07/60.03 MAYBE 202.07/60.03 202.07/60.03 The weightgap principle applies (using the following nonconstant 202.07/60.03 growth matrix-interpretation) 202.07/60.03 202.07/60.03 TcT has computed the following triangular matrix interpretation. 202.07/60.03 Note that the diagonal of the component-wise maxima of 202.07/60.03 interpretation-entries contains no more than 1 non-zero entries. 202.07/60.03 202.07/60.03 [a__f](x1) = [1] x1 + [1] 202.07/60.03 202.07/60.03 [a__if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] 202.07/60.03 202.07/60.03 [mark](x1) = [1] x1 + [0] 202.07/60.03 202.07/60.03 [c] = [0] 202.07/60.03 202.07/60.03 [f](x1) = [1] x1 + [0] 202.07/60.03 202.07/60.03 [true] = [1] 202.07/60.03 202.07/60.03 [false] = [1] 202.07/60.03 202.07/60.03 [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] 202.07/60.03 202.07/60.03 The order satisfies the following ordering constraints: 202.07/60.03 202.07/60.03 [a__f(X)] = [1] X + [1] 202.07/60.03 >= [1] X + [1] 202.07/60.03 = [a__if(mark(X), c(), f(true()))] 202.07/60.03 202.07/60.03 [a__f(X)] = [1] X + [1] 202.07/60.03 > [1] X + [0] 202.07/60.03 = [f(X)] 202.07/60.03 202.07/60.03 [a__if(X1, X2, X3)] = [1] X1 + [1] X2 + [1] X3 + [0] 202.07/60.03 >= [1] X1 + [1] X2 + [1] X3 + [0] 202.07/60.03 = [if(X1, X2, X3)] 202.07/60.03 202.07/60.03 [a__if(true(), X, Y)] = [1] X + [1] Y + [1] 202.07/60.03 > [1] X + [0] 202.07/60.03 = [mark(X)] 202.07/60.03 202.07/60.03 [a__if(false(), X, Y)] = [1] X + [1] Y + [1] 202.07/60.03 > [1] Y + [0] 202.07/60.03 = [mark(Y)] 202.07/60.03 202.07/60.03 [mark(c())] = [0] 202.07/60.03 >= [0] 202.07/60.03 = [c()] 202.07/60.03 202.07/60.03 [mark(f(X))] = [1] X + [0] 202.07/60.03 ? [1] X + [1] 202.07/60.03 = [a__f(mark(X))] 202.07/60.03 202.07/60.03 [mark(true())] = [1] 202.07/60.03 >= [1] 202.07/60.03 = [true()] 202.07/60.03 202.07/60.03 [mark(false())] = [1] 202.07/60.03 >= [1] 202.07/60.03 = [false()] 202.07/60.03 202.07/60.03 [mark(if(X1, X2, X3))] = [1] X1 + [1] X2 + [1] X3 + [0] 202.07/60.03 >= [1] X1 + [1] X2 + [1] X3 + [0] 202.07/60.03 = [a__if(mark(X1), mark(X2), X3)] 202.07/60.03 202.07/60.03 202.07/60.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 202.07/60.03 202.07/60.03 We are left with following problem, upon which TcT provides the 202.07/60.03 certificate MAYBE. 202.07/60.03 202.07/60.03 Strict Trs: 202.07/60.03 { a__if(X1, X2, X3) -> if(X1, X2, X3) 202.07/60.03 , mark(c()) -> c() 202.07/60.03 , mark(f(X)) -> a__f(mark(X)) 202.07/60.03 , mark(true()) -> true() 202.07/60.03 , mark(false()) -> false() } 202.07/60.03 Weak Trs: 202.07/60.03 { a__f(X) -> a__if(mark(X), c(), f(true())) 202.07/60.03 , a__f(X) -> f(X) 202.07/60.03 , a__if(true(), X, Y) -> mark(X) 202.07/60.03 , a__if(false(), X, Y) -> mark(Y) 202.07/60.03 , mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3) } 202.07/60.03 Obligation: 202.07/60.03 derivational complexity 202.07/60.03 Answer: 202.07/60.03 MAYBE 202.07/60.03 202.07/60.03 The weightgap principle applies (using the following nonconstant 202.07/60.03 growth matrix-interpretation) 202.07/60.03 202.07/60.03 TcT has computed the following triangular matrix interpretation. 202.07/60.03 Note that the diagonal of the component-wise maxima of 202.07/60.03 interpretation-entries contains no more than 1 non-zero entries. 202.07/60.03 202.07/60.03 [a__f](x1) = [1] x1 + [2] 202.07/60.03 202.07/60.03 [a__if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] 202.07/60.03 202.07/60.03 [mark](x1) = [1] x1 + [1] 202.07/60.03 202.07/60.03 [c] = [0] 202.07/60.03 202.07/60.03 [f](x1) = [1] x1 + [0] 202.07/60.03 202.07/60.03 [true] = [1] 202.07/60.03 202.07/60.03 [false] = [1] 202.07/60.03 202.07/60.03 [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [2] 202.07/60.03 202.07/60.03 The order satisfies the following ordering constraints: 202.07/60.03 202.07/60.03 [a__f(X)] = [1] X + [2] 202.07/60.03 >= [1] X + [2] 202.07/60.03 = [a__if(mark(X), c(), f(true()))] 202.07/60.03 202.07/60.03 [a__f(X)] = [1] X + [2] 202.07/60.03 > [1] X + [0] 202.07/60.03 = [f(X)] 202.07/60.03 202.07/60.03 [a__if(X1, X2, X3)] = [1] X1 + [1] X2 + [1] X3 + [0] 202.07/60.03 ? [1] X1 + [1] X2 + [1] X3 + [2] 202.07/60.03 = [if(X1, X2, X3)] 202.07/60.03 202.07/60.03 [a__if(true(), X, Y)] = [1] X + [1] Y + [1] 202.07/60.03 >= [1] X + [1] 202.07/60.03 = [mark(X)] 202.07/60.03 202.07/60.03 [a__if(false(), X, Y)] = [1] X + [1] Y + [1] 202.07/60.03 >= [1] Y + [1] 202.07/60.03 = [mark(Y)] 202.07/60.03 202.07/60.03 [mark(c())] = [1] 202.07/60.03 > [0] 202.07/60.03 = [c()] 202.07/60.03 202.07/60.03 [mark(f(X))] = [1] X + [1] 202.07/60.03 ? [1] X + [3] 202.07/60.03 = [a__f(mark(X))] 202.07/60.03 202.07/60.03 [mark(true())] = [2] 202.07/60.03 > [1] 202.07/60.03 = [true()] 202.07/60.03 202.07/60.03 [mark(false())] = [2] 202.07/60.03 > [1] 202.07/60.03 = [false()] 202.07/60.03 202.07/60.03 [mark(if(X1, X2, X3))] = [1] X1 + [1] X2 + [1] X3 + [3] 202.07/60.03 > [1] X1 + [1] X2 + [1] X3 + [2] 202.07/60.03 = [a__if(mark(X1), mark(X2), X3)] 202.07/60.03 202.07/60.03 202.07/60.03 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 202.07/60.03 202.07/60.03 We are left with following problem, upon which TcT provides the 202.07/60.03 certificate MAYBE. 202.07/60.03 202.07/60.03 Strict Trs: 202.07/60.03 { a__if(X1, X2, X3) -> if(X1, X2, X3) 202.07/60.03 , mark(f(X)) -> a__f(mark(X)) } 202.07/60.03 Weak Trs: 202.07/60.03 { a__f(X) -> a__if(mark(X), c(), f(true())) 202.07/60.03 , a__f(X) -> f(X) 202.07/60.03 , a__if(true(), X, Y) -> mark(X) 202.07/60.03 , a__if(false(), X, Y) -> mark(Y) 202.07/60.03 , mark(c()) -> c() 202.07/60.03 , mark(true()) -> true() 202.07/60.03 , mark(false()) -> false() 202.07/60.03 , mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3) } 202.07/60.03 Obligation: 202.07/60.03 derivational complexity 202.07/60.03 Answer: 202.07/60.03 MAYBE 202.07/60.03 202.07/60.03 None of the processors succeeded. 202.07/60.03 202.07/60.03 Details of failed attempt(s): 202.07/60.03 ----------------------------- 202.07/60.03 1) 'empty' failed due to the following reason: 202.07/60.03 202.07/60.03 Empty strict component of the problem is NOT empty. 202.07/60.03 202.07/60.03 2) 'Fastest' failed due to the following reason: 202.07/60.03 202.07/60.03 None of the processors succeeded. 202.07/60.03 202.07/60.03 Details of failed attempt(s): 202.07/60.03 ----------------------------- 202.07/60.03 1) 'Fastest (timeout of 30 seconds)' failed due to the following 202.07/60.03 reason: 202.07/60.03 202.07/60.03 Computation stopped due to timeout after 30.0 seconds. 202.07/60.03 202.07/60.03 2) 'Fastest' failed due to the following reason: 202.07/60.03 202.07/60.03 None of the processors succeeded. 202.07/60.03 202.07/60.03 Details of failed attempt(s): 202.07/60.03 ----------------------------- 202.07/60.03 1) 'matrix interpretation of dimension 6' failed due to the 202.07/60.03 following reason: 202.07/60.03 202.07/60.03 The input cannot be shown compatible 202.07/60.03 202.07/60.03 2) 'matrix interpretation of dimension 5' failed due to the 202.07/60.03 following reason: 202.07/60.03 202.07/60.03 The input cannot be shown compatible 202.07/60.03 202.07/60.03 3) 'matrix interpretation of dimension 4' failed due to the 202.07/60.03 following reason: 202.07/60.03 202.07/60.03 The input cannot be shown compatible 202.07/60.03 202.07/60.03 4) 'matrix interpretation of dimension 3' failed due to the 202.07/60.03 following reason: 202.07/60.03 202.07/60.03 The input cannot be shown compatible 202.07/60.03 202.07/60.03 5) 'matrix interpretation of dimension 2' failed due to the 202.07/60.03 following reason: 202.07/60.03 202.07/60.03 The input cannot be shown compatible 202.07/60.03 202.07/60.03 6) 'matrix interpretation of dimension 1' failed due to the 202.07/60.03 following reason: 202.07/60.03 202.07/60.03 The input cannot be shown compatible 202.07/60.03 202.07/60.03 202.07/60.03 3) 'iteProgress' failed due to the following reason: 202.07/60.03 202.07/60.03 Fail 202.07/60.03 202.07/60.03 4) 'bsearch-matrix' failed due to the following reason: 202.07/60.03 202.07/60.03 The input cannot be shown compatible 202.07/60.03 202.07/60.03 202.07/60.03 202.07/60.03 3) 'iteProgress (timeout of 297 seconds)' failed due to the 202.07/60.03 following reason: 202.07/60.03 202.07/60.03 Fail 202.07/60.03 202.07/60.03 4) 'bsearch-matrix (timeout of 297 seconds)' failed due to the 202.07/60.03 following reason: 202.07/60.03 202.07/60.03 The input cannot be shown compatible 202.07/60.03 202.07/60.03 202.07/60.03 Arrrr.. 202.07/60.07 EOF