YES(O(1),O(n^2)) 219.81/61.10 YES(O(1),O(n^2)) 219.81/61.10 219.81/61.10 We are left with following problem, upon which TcT provides the 219.81/61.10 certificate YES(O(1),O(n^2)). 219.81/61.10 219.81/61.10 Strict Trs: 219.81/61.10 { active(g(X)) -> mark(h(X)) 219.81/61.10 , active(h(d())) -> mark(g(c())) 219.81/61.10 , active(c()) -> mark(d()) 219.81/61.10 , g(ok(X)) -> ok(g(X)) 219.81/61.10 , h(ok(X)) -> ok(h(X)) 219.81/61.10 , proper(g(X)) -> g(proper(X)) 219.81/61.10 , proper(h(X)) -> h(proper(X)) 219.81/61.10 , proper(c()) -> ok(c()) 219.81/61.10 , proper(d()) -> ok(d()) 219.81/61.10 , top(mark(X)) -> top(proper(X)) 219.81/61.10 , top(ok(X)) -> top(active(X)) } 219.81/61.10 Obligation: 219.81/61.10 derivational complexity 219.81/61.10 Answer: 219.81/61.10 YES(O(1),O(n^2)) 219.81/61.10 219.81/61.10 The weightgap principle applies (using the following nonconstant 219.81/61.10 growth matrix-interpretation) 219.81/61.10 219.81/61.10 TcT has computed the following triangular matrix interpretation. 219.81/61.10 Note that the diagonal of the component-wise maxima of 219.81/61.10 interpretation-entries contains no more than 1 non-zero entries. 219.81/61.10 219.81/61.10 [active](x1) = [1] x1 + [1] 219.81/61.10 219.81/61.10 [g](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 [mark](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 [h](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 [c] = [0] 219.81/61.10 219.81/61.10 [d] = [0] 219.81/61.10 219.81/61.10 [proper](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 [ok](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 [top](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 The order satisfies the following ordering constraints: 219.81/61.10 219.81/61.10 [active(g(X))] = [1] X + [1] 219.81/61.10 > [1] X + [0] 219.81/61.10 = [mark(h(X))] 219.81/61.10 219.81/61.10 [active(h(d()))] = [1] 219.81/61.10 > [0] 219.81/61.10 = [mark(g(c()))] 219.81/61.10 219.81/61.10 [active(c())] = [1] 219.81/61.10 > [0] 219.81/61.10 = [mark(d())] 219.81/61.10 219.81/61.10 [g(ok(X))] = [1] X + [0] 219.81/61.10 >= [1] X + [0] 219.81/61.10 = [ok(g(X))] 219.81/61.10 219.81/61.10 [h(ok(X))] = [1] X + [0] 219.81/61.10 >= [1] X + [0] 219.81/61.10 = [ok(h(X))] 219.81/61.10 219.81/61.10 [proper(g(X))] = [1] X + [0] 219.81/61.10 >= [1] X + [0] 219.81/61.10 = [g(proper(X))] 219.81/61.10 219.81/61.10 [proper(h(X))] = [1] X + [0] 219.81/61.10 >= [1] X + [0] 219.81/61.10 = [h(proper(X))] 219.81/61.10 219.81/61.10 [proper(c())] = [0] 219.81/61.10 >= [0] 219.81/61.10 = [ok(c())] 219.81/61.10 219.81/61.10 [proper(d())] = [0] 219.81/61.10 >= [0] 219.81/61.10 = [ok(d())] 219.81/61.10 219.81/61.10 [top(mark(X))] = [1] X + [0] 219.81/61.10 >= [1] X + [0] 219.81/61.10 = [top(proper(X))] 219.81/61.10 219.81/61.10 [top(ok(X))] = [1] X + [0] 219.81/61.10 ? [1] X + [1] 219.81/61.10 = [top(active(X))] 219.81/61.10 219.81/61.10 219.81/61.10 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 219.81/61.10 219.81/61.10 We are left with following problem, upon which TcT provides the 219.81/61.10 certificate YES(O(1),O(n^2)). 219.81/61.10 219.81/61.10 Strict Trs: 219.81/61.10 { g(ok(X)) -> ok(g(X)) 219.81/61.10 , h(ok(X)) -> ok(h(X)) 219.81/61.10 , proper(g(X)) -> g(proper(X)) 219.81/61.10 , proper(h(X)) -> h(proper(X)) 219.81/61.10 , proper(c()) -> ok(c()) 219.81/61.10 , proper(d()) -> ok(d()) 219.81/61.10 , top(mark(X)) -> top(proper(X)) 219.81/61.10 , top(ok(X)) -> top(active(X)) } 219.81/61.10 Weak Trs: 219.81/61.10 { active(g(X)) -> mark(h(X)) 219.81/61.10 , active(h(d())) -> mark(g(c())) 219.81/61.10 , active(c()) -> mark(d()) } 219.81/61.10 Obligation: 219.81/61.10 derivational complexity 219.81/61.10 Answer: 219.81/61.10 YES(O(1),O(n^2)) 219.81/61.10 219.81/61.10 The weightgap principle applies (using the following nonconstant 219.81/61.10 growth matrix-interpretation) 219.81/61.10 219.81/61.10 TcT has computed the following triangular matrix interpretation. 219.81/61.10 Note that the diagonal of the component-wise maxima of 219.81/61.10 interpretation-entries contains no more than 1 non-zero entries. 219.81/61.10 219.81/61.10 [active](x1) = [1] x1 + [1] 219.81/61.10 219.81/61.10 [g](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 [mark](x1) = [1] x1 + [1] 219.81/61.10 219.81/61.10 [h](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 [c] = [0] 219.81/61.10 219.81/61.10 [d] = [0] 219.81/61.10 219.81/61.10 [proper](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 [ok](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 [top](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 The order satisfies the following ordering constraints: 219.81/61.10 219.81/61.10 [active(g(X))] = [1] X + [1] 219.81/61.10 >= [1] X + [1] 219.81/61.10 = [mark(h(X))] 219.81/61.10 219.81/61.10 [active(h(d()))] = [1] 219.81/61.10 >= [1] 219.81/61.10 = [mark(g(c()))] 219.81/61.10 219.81/61.10 [active(c())] = [1] 219.81/61.10 >= [1] 219.81/61.10 = [mark(d())] 219.81/61.10 219.81/61.10 [g(ok(X))] = [1] X + [0] 219.81/61.10 >= [1] X + [0] 219.81/61.10 = [ok(g(X))] 219.81/61.10 219.81/61.10 [h(ok(X))] = [1] X + [0] 219.81/61.10 >= [1] X + [0] 219.81/61.10 = [ok(h(X))] 219.81/61.10 219.81/61.10 [proper(g(X))] = [1] X + [0] 219.81/61.10 >= [1] X + [0] 219.81/61.10 = [g(proper(X))] 219.81/61.10 219.81/61.10 [proper(h(X))] = [1] X + [0] 219.81/61.10 >= [1] X + [0] 219.81/61.10 = [h(proper(X))] 219.81/61.10 219.81/61.10 [proper(c())] = [0] 219.81/61.10 >= [0] 219.81/61.10 = [ok(c())] 219.81/61.10 219.81/61.10 [proper(d())] = [0] 219.81/61.10 >= [0] 219.81/61.10 = [ok(d())] 219.81/61.10 219.81/61.10 [top(mark(X))] = [1] X + [1] 219.81/61.10 > [1] X + [0] 219.81/61.10 = [top(proper(X))] 219.81/61.10 219.81/61.10 [top(ok(X))] = [1] X + [0] 219.81/61.10 ? [1] X + [1] 219.81/61.10 = [top(active(X))] 219.81/61.10 219.81/61.10 219.81/61.10 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 219.81/61.10 219.81/61.10 We are left with following problem, upon which TcT provides the 219.81/61.10 certificate YES(O(1),O(n^2)). 219.81/61.10 219.81/61.10 Strict Trs: 219.81/61.10 { g(ok(X)) -> ok(g(X)) 219.81/61.10 , h(ok(X)) -> ok(h(X)) 219.81/61.10 , proper(g(X)) -> g(proper(X)) 219.81/61.10 , proper(h(X)) -> h(proper(X)) 219.81/61.10 , proper(c()) -> ok(c()) 219.81/61.10 , proper(d()) -> ok(d()) 219.81/61.10 , top(ok(X)) -> top(active(X)) } 219.81/61.10 Weak Trs: 219.81/61.10 { active(g(X)) -> mark(h(X)) 219.81/61.10 , active(h(d())) -> mark(g(c())) 219.81/61.10 , active(c()) -> mark(d()) 219.81/61.10 , top(mark(X)) -> top(proper(X)) } 219.81/61.10 Obligation: 219.81/61.10 derivational complexity 219.81/61.10 Answer: 219.81/61.10 YES(O(1),O(n^2)) 219.81/61.10 219.81/61.10 The weightgap principle applies (using the following nonconstant 219.81/61.10 growth matrix-interpretation) 219.81/61.10 219.81/61.10 TcT has computed the following triangular matrix interpretation. 219.81/61.10 Note that the diagonal of the component-wise maxima of 219.81/61.10 interpretation-entries contains no more than 1 non-zero entries. 219.81/61.10 219.81/61.10 [active](x1) = [1] x1 + [1] 219.81/61.10 219.81/61.10 [g](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 [mark](x1) = [1] x1 + [1] 219.81/61.10 219.81/61.10 [h](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 [c] = [0] 219.81/61.10 219.81/61.10 [d] = [0] 219.81/61.10 219.81/61.10 [proper](x1) = [1] x1 + [1] 219.81/61.10 219.81/61.10 [ok](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 [top](x1) = [1] x1 + [0] 219.81/61.10 219.81/61.10 The order satisfies the following ordering constraints: 219.81/61.10 219.81/61.10 [active(g(X))] = [1] X + [1] 219.81/61.10 >= [1] X + [1] 219.81/61.10 = [mark(h(X))] 219.81/61.10 219.81/61.10 [active(h(d()))] = [1] 219.81/61.10 >= [1] 219.81/61.10 = [mark(g(c()))] 219.81/61.10 219.81/61.10 [active(c())] = [1] 219.81/61.10 >= [1] 219.81/61.10 = [mark(d())] 219.81/61.10 219.81/61.10 [g(ok(X))] = [1] X + [0] 219.81/61.10 >= [1] X + [0] 219.81/61.10 = [ok(g(X))] 219.81/61.10 219.81/61.10 [h(ok(X))] = [1] X + [0] 219.81/61.10 >= [1] X + [0] 219.81/61.10 = [ok(h(X))] 219.81/61.10 219.81/61.10 [proper(g(X))] = [1] X + [1] 219.81/61.10 >= [1] X + [1] 219.81/61.10 = [g(proper(X))] 219.81/61.10 219.81/61.10 [proper(h(X))] = [1] X + [1] 219.81/61.10 >= [1] X + [1] 219.81/61.10 = [h(proper(X))] 219.81/61.10 219.81/61.10 [proper(c())] = [1] 219.81/61.10 > [0] 219.81/61.10 = [ok(c())] 219.81/61.10 219.81/61.10 [proper(d())] = [1] 219.81/61.10 > [0] 219.81/61.10 = [ok(d())] 219.81/61.10 219.81/61.10 [top(mark(X))] = [1] X + [1] 219.81/61.10 >= [1] X + [1] 219.81/61.10 = [top(proper(X))] 219.81/61.10 219.81/61.10 [top(ok(X))] = [1] X + [0] 219.81/61.10 ? [1] X + [1] 219.81/61.10 = [top(active(X))] 219.81/61.10 219.81/61.10 219.81/61.10 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 219.81/61.10 219.81/61.10 We are left with following problem, upon which TcT provides the 219.81/61.10 certificate YES(O(1),O(n^2)). 219.81/61.10 219.81/61.10 Strict Trs: 219.81/61.10 { g(ok(X)) -> ok(g(X)) 219.81/61.10 , h(ok(X)) -> ok(h(X)) 219.81/61.10 , proper(g(X)) -> g(proper(X)) 219.81/61.10 , proper(h(X)) -> h(proper(X)) 219.81/61.10 , top(ok(X)) -> top(active(X)) } 219.81/61.10 Weak Trs: 219.81/61.10 { active(g(X)) -> mark(h(X)) 219.81/61.10 , active(h(d())) -> mark(g(c())) 219.81/61.10 , active(c()) -> mark(d()) 219.81/61.10 , proper(c()) -> ok(c()) 219.81/61.10 , proper(d()) -> ok(d()) 219.81/61.10 , top(mark(X)) -> top(proper(X)) } 219.81/61.10 Obligation: 219.81/61.10 derivational complexity 219.81/61.10 Answer: 219.81/61.10 YES(O(1),O(n^2)) 219.81/61.10 219.81/61.10 We use the processor 'matrix interpretation of dimension 3' to 219.81/61.10 orient following rules strictly. 219.81/61.10 219.81/61.10 Trs: { top(ok(X)) -> top(active(X)) } 219.81/61.10 219.81/61.10 The induced complexity on above rules (modulo remaining rules) is 219.81/61.10 YES(?,O(n^1)) . These rules are moved into the corresponding weak 219.81/61.10 component(s). 219.81/61.10 219.81/61.10 Sub-proof: 219.81/61.10 ---------- 219.81/61.10 TcT has computed the following triangular matrix interpretation. 219.81/61.10 Note that the diagonal of the component-wise maxima of 219.81/61.10 interpretation-entries contains no more than 1 non-zero entries. 219.81/61.10 219.81/61.10 [1 0 2] [1] 219.81/61.10 [active](x1) = [0 0 1] x1 + [0] 219.81/61.10 [0 0 0] [0] 219.81/61.10 219.81/61.10 [1 0 0] [1] 219.81/61.10 [g](x1) = [0 0 2] x1 + [0] 219.81/61.10 [0 0 0] [1] 219.81/61.10 219.81/61.10 [1 0 2] [2] 219.81/61.10 [mark](x1) = [0 0 1] x1 + [0] 219.81/61.10 [0 0 0] [0] 219.81/61.10 219.81/61.10 [1 0 0] [0] 219.81/61.10 [h](x1) = [0 0 1] x1 + [1] 219.81/61.10 [0 0 0] [1] 219.81/61.10 219.81/61.10 [0] 219.81/61.10 [c] = [0] 219.81/61.10 [2] 219.81/61.10 219.81/61.10 [2] 219.81/61.10 [d] = [0] 219.81/61.10 [0] 219.81/61.10 219.81/61.10 [1 0 0] [0] 219.81/61.10 [proper](x1) = [0 0 2] x1 + [0] 219.81/61.10 [0 0 0] [1] 219.81/61.10 219.81/61.10 [1 0 0] [0] 219.81/61.10 [ok](x1) = [0 0 2] x1 + [0] 219.81/61.10 [0 0 0] [1] 219.81/61.10 219.81/61.10 [1 2 2] [0] 219.81/61.10 [top](x1) = [0 0 0] x1 + [0] 219.81/61.10 [0 0 0] [1] 219.81/61.10 219.81/61.10 The order satisfies the following ordering constraints: 219.81/61.10 219.81/61.10 [active(g(X))] = [1 0 0] [4] 219.81/61.10 [0 0 0] X + [1] 219.81/61.10 [0 0 0] [0] 219.81/61.10 >= [1 0 0] [4] 219.81/61.10 [0 0 0] X + [1] 219.81/61.10 [0 0 0] [0] 219.81/61.10 = [mark(h(X))] 219.81/61.10 219.81/61.10 [active(h(d()))] = [5] 219.81/61.10 [1] 219.81/61.10 [0] 219.81/61.10 >= [5] 219.81/61.10 [1] 219.81/61.10 [0] 219.81/61.10 = [mark(g(c()))] 219.81/61.10 219.81/61.10 [active(c())] = [5] 219.81/61.10 [2] 219.81/61.10 [0] 219.81/61.10 > [4] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 = [mark(d())] 219.81/61.10 219.81/61.10 [g(ok(X))] = [1 0 0] [1] 219.81/61.10 [0 0 0] X + [2] 219.81/61.10 [0 0 0] [1] 219.81/61.10 >= [1 0 0] [1] 219.81/61.10 [0 0 0] X + [2] 219.81/61.10 [0 0 0] [1] 219.81/61.10 = [ok(g(X))] 219.81/61.10 219.81/61.10 [h(ok(X))] = [1 0 0] [0] 219.81/61.10 [0 0 0] X + [2] 219.81/61.10 [0 0 0] [1] 219.81/61.10 >= [1 0 0] [0] 219.81/61.10 [0 0 0] X + [2] 219.81/61.10 [0 0 0] [1] 219.81/61.10 = [ok(h(X))] 219.81/61.10 219.81/61.10 [proper(g(X))] = [1 0 0] [1] 219.81/61.10 [0 0 0] X + [2] 219.81/61.10 [0 0 0] [1] 219.81/61.10 >= [1 0 0] [1] 219.81/61.10 [0 0 0] X + [2] 219.81/61.10 [0 0 0] [1] 219.81/61.10 = [g(proper(X))] 219.81/61.10 219.81/61.10 [proper(h(X))] = [1 0 0] [0] 219.81/61.10 [0 0 0] X + [2] 219.81/61.10 [0 0 0] [1] 219.81/61.10 >= [1 0 0] [0] 219.81/61.10 [0 0 0] X + [2] 219.81/61.10 [0 0 0] [1] 219.81/61.10 = [h(proper(X))] 219.81/61.10 219.81/61.10 [proper(c())] = [0] 219.81/61.10 [4] 219.81/61.10 [1] 219.81/61.10 >= [0] 219.81/61.10 [4] 219.81/61.10 [1] 219.81/61.10 = [ok(c())] 219.81/61.10 219.81/61.10 [proper(d())] = [2] 219.81/61.10 [0] 219.81/61.10 [1] 219.81/61.10 >= [2] 219.81/61.10 [0] 219.81/61.10 [1] 219.81/61.10 = [ok(d())] 219.81/61.10 219.81/61.10 [top(mark(X))] = [1 0 4] [2] 219.81/61.10 [0 0 0] X + [0] 219.81/61.10 [0 0 0] [1] 219.81/61.10 >= [1 0 4] [2] 219.81/61.10 [0 0 0] X + [0] 219.81/61.10 [0 0 0] [1] 219.81/61.10 = [top(proper(X))] 219.81/61.10 219.81/61.10 [top(ok(X))] = [1 0 4] [2] 219.81/61.10 [0 0 0] X + [0] 219.81/61.10 [0 0 0] [1] 219.81/61.10 > [1 0 4] [1] 219.81/61.10 [0 0 0] X + [0] 219.81/61.10 [0 0 0] [1] 219.81/61.10 = [top(active(X))] 219.81/61.10 219.81/61.10 219.81/61.10 We return to the main proof. 219.81/61.10 219.81/61.10 We are left with following problem, upon which TcT provides the 219.81/61.10 certificate YES(O(1),O(n^2)). 219.81/61.10 219.81/61.10 Strict Trs: 219.81/61.10 { g(ok(X)) -> ok(g(X)) 219.81/61.10 , h(ok(X)) -> ok(h(X)) 219.81/61.10 , proper(g(X)) -> g(proper(X)) 219.81/61.10 , proper(h(X)) -> h(proper(X)) } 219.81/61.10 Weak Trs: 219.81/61.10 { active(g(X)) -> mark(h(X)) 219.81/61.10 , active(h(d())) -> mark(g(c())) 219.81/61.10 , active(c()) -> mark(d()) 219.81/61.10 , proper(c()) -> ok(c()) 219.81/61.10 , proper(d()) -> ok(d()) 219.81/61.10 , top(mark(X)) -> top(proper(X)) 219.81/61.10 , top(ok(X)) -> top(active(X)) } 219.81/61.10 Obligation: 219.81/61.10 derivational complexity 219.81/61.10 Answer: 219.81/61.10 YES(O(1),O(n^2)) 219.81/61.10 219.81/61.10 We use the processor 'matrix interpretation of dimension 4' to 219.81/61.10 orient following rules strictly. 219.81/61.10 219.81/61.10 Trs: { proper(h(X)) -> h(proper(X)) } 219.81/61.10 219.81/61.10 The induced complexity on above rules (modulo remaining rules) is 219.81/61.10 YES(?,O(n^2)) . These rules are moved into the corresponding weak 219.81/61.10 component(s). 219.81/61.10 219.81/61.10 Sub-proof: 219.81/61.10 ---------- 219.81/61.10 TcT has computed the following triangular matrix interpretation. 219.81/61.10 Note that the diagonal of the component-wise maxima of 219.81/61.10 interpretation-entries contains no more than 2 non-zero entries. 219.81/61.10 219.81/61.10 [1 0 1 0] [0] 219.81/61.10 [active](x1) = [0 0 0 0] x1 + [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 219.81/61.10 [1 0 0 0] [1] 219.81/61.10 [g](x1) = [0 0 0 1] x1 + [0] 219.81/61.10 [0 0 0 1] [0] 219.81/61.10 [0 0 0 1] [0] 219.81/61.10 219.81/61.10 [1 0 0 1] [0] 219.81/61.10 [mark](x1) = [0 0 1 0] x1 + [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 219.81/61.10 [1 0 0 0] [0] 219.81/61.10 [h](x1) = [0 0 0 0] x1 + [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 [0 0 0 1] [1] 219.81/61.10 219.81/61.10 [0] 219.81/61.10 [c] = [0] 219.81/61.10 [1] 219.81/61.10 [0] 219.81/61.10 219.81/61.10 [1] 219.81/61.10 [d] = [0] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 219.81/61.10 [1 0 0 1] [0] 219.81/61.10 [proper](x1) = [0 0 1 0] x1 + [0] 219.81/61.10 [0 0 0 1] [0] 219.81/61.10 [0 0 0 1] [0] 219.81/61.10 219.81/61.10 [1 0 0 0] [0] 219.81/61.10 [ok](x1) = [0 0 1 0] x1 + [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 [0 0 0 1] [0] 219.81/61.10 219.81/61.10 [1 1 0 0] [0] 219.81/61.10 [top](x1) = [0 0 0 0] x1 + [0] 219.81/61.10 [0 0 0 0] [1] 219.81/61.10 [0 0 0 0] [1] 219.81/61.10 219.81/61.10 The order satisfies the following ordering constraints: 219.81/61.10 219.81/61.10 [active(g(X))] = [1 0 0 1] [1] 219.81/61.10 [0 0 0 0] X + [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 >= [1 0 0 1] [1] 219.81/61.10 [0 0 0 0] X + [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 = [mark(h(X))] 219.81/61.10 219.81/61.10 [active(h(d()))] = [1] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 >= [1] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 = [mark(g(c()))] 219.81/61.10 219.81/61.10 [active(c())] = [1] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 >= [1] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 = [mark(d())] 219.81/61.10 219.81/61.10 [g(ok(X))] = [1 0 0 0] [1] 219.81/61.10 [0 0 0 1] X + [0] 219.81/61.10 [0 0 0 1] [0] 219.81/61.10 [0 0 0 1] [0] 219.81/61.10 >= [1 0 0 0] [1] 219.81/61.10 [0 0 0 1] X + [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 [0 0 0 1] [0] 219.81/61.10 = [ok(g(X))] 219.81/61.10 219.81/61.10 [h(ok(X))] = [1 0 0 0] [0] 219.81/61.10 [0 0 0 0] X + [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 [0 0 0 1] [1] 219.81/61.10 >= [1 0 0 0] [0] 219.81/61.10 [0 0 0 0] X + [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 [0 0 0 1] [1] 219.81/61.10 = [ok(h(X))] 219.81/61.10 219.81/61.10 [proper(g(X))] = [1 0 0 1] [1] 219.81/61.10 [0 0 0 1] X + [0] 219.81/61.10 [0 0 0 1] [0] 219.81/61.10 [0 0 0 1] [0] 219.81/61.10 >= [1 0 0 1] [1] 219.81/61.10 [0 0 0 1] X + [0] 219.81/61.10 [0 0 0 1] [0] 219.81/61.10 [0 0 0 1] [0] 219.81/61.10 = [g(proper(X))] 219.81/61.10 219.81/61.10 [proper(h(X))] = [1 0 0 1] [1] 219.81/61.10 [0 0 0 0] X + [0] 219.81/61.10 [0 0 0 1] [1] 219.81/61.10 [0 0 0 1] [1] 219.81/61.10 > [1 0 0 1] [0] 219.81/61.10 [0 0 0 0] X + [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 [0 0 0 1] [1] 219.81/61.10 = [h(proper(X))] 219.81/61.10 219.81/61.10 [proper(c())] = [0] 219.81/61.10 [1] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 >= [0] 219.81/61.10 [1] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 = [ok(c())] 219.81/61.10 219.81/61.10 [proper(d())] = [1] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 >= [1] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 [0] 219.81/61.10 = [ok(d())] 219.81/61.10 219.81/61.10 [top(mark(X))] = [1 0 1 1] [0] 219.81/61.10 [0 0 0 0] X + [0] 219.81/61.10 [0 0 0 0] [1] 219.81/61.10 [0 0 0 0] [1] 219.81/61.10 >= [1 0 1 1] [0] 219.81/61.10 [0 0 0 0] X + [0] 219.81/61.10 [0 0 0 0] [1] 219.81/61.10 [0 0 0 0] [1] 219.81/61.10 = [top(proper(X))] 219.81/61.10 219.81/61.10 [top(ok(X))] = [1 0 1 0] [0] 219.81/61.10 [0 0 0 0] X + [0] 219.81/61.10 [0 0 0 0] [1] 219.81/61.10 [0 0 0 0] [1] 219.81/61.10 >= [1 0 1 0] [0] 219.81/61.10 [0 0 0 0] X + [0] 219.81/61.10 [0 0 0 0] [1] 219.81/61.10 [0 0 0 0] [1] 219.81/61.10 = [top(active(X))] 219.81/61.10 219.81/61.10 219.81/61.10 We return to the main proof. 219.81/61.10 219.81/61.10 We are left with following problem, upon which TcT provides the 219.81/61.10 certificate YES(O(1),O(n^2)). 219.81/61.10 219.81/61.10 Strict Trs: 219.81/61.10 { g(ok(X)) -> ok(g(X)) 219.81/61.10 , h(ok(X)) -> ok(h(X)) 219.81/61.10 , proper(g(X)) -> g(proper(X)) } 219.81/61.10 Weak Trs: 219.81/61.10 { active(g(X)) -> mark(h(X)) 219.81/61.10 , active(h(d())) -> mark(g(c())) 219.81/61.10 , active(c()) -> mark(d()) 219.81/61.10 , proper(h(X)) -> h(proper(X)) 219.81/61.10 , proper(c()) -> ok(c()) 219.81/61.10 , proper(d()) -> ok(d()) 219.81/61.10 , top(mark(X)) -> top(proper(X)) 219.81/61.10 , top(ok(X)) -> top(active(X)) } 219.81/61.10 Obligation: 219.81/61.10 derivational complexity 219.81/61.10 Answer: 219.81/61.10 YES(O(1),O(n^2)) 219.81/61.10 219.81/61.10 We use the processor 'matrix interpretation of dimension 4' to 219.81/61.10 orient following rules strictly. 219.81/61.10 219.81/61.10 Trs: { proper(g(X)) -> g(proper(X)) } 219.81/61.10 219.81/61.10 The induced complexity on above rules (modulo remaining rules) is 219.81/61.10 YES(?,O(n^2)) . These rules are moved into the corresponding weak 219.81/61.10 component(s). 219.81/61.10 219.81/61.10 Sub-proof: 219.81/61.10 ---------- 219.81/61.10 TcT has computed the following triangular matrix interpretation. 219.81/61.10 Note that the diagonal of the component-wise maxima of 219.81/61.10 interpretation-entries contains no more than 2 non-zero entries. 219.81/61.10 219.81/61.10 [1 0 0 1] [0] 219.81/61.10 [active](x1) = [0 0 0 0] x1 + [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 219.81/61.10 [1 1 0 0] [0] 219.81/61.10 [g](x1) = [0 1 0 0] x1 + [1] 219.81/61.10 [0 0 0 0] [0] 219.81/61.10 [0 0 0 0] [0] 219.81/61.11 219.81/61.11 [1 1 0 0] [0] 219.81/61.11 [mark](x1) = [0 0 0 0] x1 + [0] 219.81/61.11 [0 0 0 1] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 219.81/61.11 [1 0 0 0] [0] 219.81/61.11 [h](x1) = [0 1 0 0] x1 + [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 219.81/61.11 [0] 219.81/61.11 [c] = [0] 219.81/61.11 [0] 219.81/61.11 [1] 219.81/61.11 219.81/61.11 [1] 219.81/61.11 [d] = [0] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 219.81/61.11 [1 1 0 0] [0] 219.81/61.11 [proper](x1) = [0 1 0 0] x1 + [0] 219.81/61.11 [0 0 0 1] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 219.81/61.11 [1 0 0 0] [0] 219.81/61.11 [ok](x1) = [0 1 0 0] x1 + [0] 219.81/61.11 [0 0 0 1] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 219.81/61.11 [1 0 1 0] [0] 219.81/61.11 [top](x1) = [0 0 0 0] x1 + [0] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 219.81/61.11 The order satisfies the following ordering constraints: 219.81/61.11 219.81/61.11 [active(g(X))] = [1 1 0 0] [0] 219.81/61.11 [0 0 0 0] X + [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 >= [1 1 0 0] [0] 219.81/61.11 [0 0 0 0] X + [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 = [mark(h(X))] 219.81/61.11 219.81/61.11 [active(h(d()))] = [1] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 >= [1] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 = [mark(g(c()))] 219.81/61.11 219.81/61.11 [active(c())] = [1] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 >= [1] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 = [mark(d())] 219.81/61.11 219.81/61.11 [g(ok(X))] = [1 1 0 0] [0] 219.81/61.11 [0 1 0 0] X + [1] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 >= [1 1 0 0] [0] 219.81/61.11 [0 1 0 0] X + [1] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 = [ok(g(X))] 219.81/61.11 219.81/61.11 [h(ok(X))] = [1 0 0 0] [0] 219.81/61.11 [0 1 0 0] X + [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 >= [1 0 0 0] [0] 219.81/61.11 [0 1 0 0] X + [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 = [ok(h(X))] 219.81/61.11 219.81/61.11 [proper(g(X))] = [1 2 0 0] [1] 219.81/61.11 [0 1 0 0] X + [1] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 > [1 2 0 0] [0] 219.81/61.11 [0 1 0 0] X + [1] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 = [g(proper(X))] 219.81/61.11 219.81/61.11 [proper(h(X))] = [1 1 0 0] [0] 219.81/61.11 [0 1 0 0] X + [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 >= [1 1 0 0] [0] 219.81/61.11 [0 1 0 0] X + [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 = [h(proper(X))] 219.81/61.11 219.81/61.11 [proper(c())] = [0] 219.81/61.11 [0] 219.81/61.11 [1] 219.81/61.11 [0] 219.81/61.11 >= [0] 219.81/61.11 [0] 219.81/61.11 [1] 219.81/61.11 [0] 219.81/61.11 = [ok(c())] 219.81/61.11 219.81/61.11 [proper(d())] = [1] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 >= [1] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 = [ok(d())] 219.81/61.11 219.81/61.11 [top(mark(X))] = [1 1 0 1] [0] 219.81/61.11 [0 0 0 0] X + [0] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 >= [1 1 0 1] [0] 219.81/61.11 [0 0 0 0] X + [0] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 = [top(proper(X))] 219.81/61.11 219.81/61.11 [top(ok(X))] = [1 0 0 1] [0] 219.81/61.11 [0 0 0 0] X + [0] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 >= [1 0 0 1] [0] 219.81/61.11 [0 0 0 0] X + [0] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 = [top(active(X))] 219.81/61.11 219.81/61.11 219.81/61.11 We return to the main proof. 219.81/61.11 219.81/61.11 We are left with following problem, upon which TcT provides the 219.81/61.11 certificate YES(O(1),O(n^2)). 219.81/61.11 219.81/61.11 Strict Trs: 219.81/61.11 { g(ok(X)) -> ok(g(X)) 219.81/61.11 , h(ok(X)) -> ok(h(X)) } 219.81/61.11 Weak Trs: 219.81/61.11 { active(g(X)) -> mark(h(X)) 219.81/61.11 , active(h(d())) -> mark(g(c())) 219.81/61.11 , active(c()) -> mark(d()) 219.81/61.11 , proper(g(X)) -> g(proper(X)) 219.81/61.11 , proper(h(X)) -> h(proper(X)) 219.81/61.11 , proper(c()) -> ok(c()) 219.81/61.11 , proper(d()) -> ok(d()) 219.81/61.11 , top(mark(X)) -> top(proper(X)) 219.81/61.11 , top(ok(X)) -> top(active(X)) } 219.81/61.11 Obligation: 219.81/61.11 derivational complexity 219.81/61.11 Answer: 219.81/61.11 YES(O(1),O(n^2)) 219.81/61.11 219.81/61.11 We use the processor 'matrix interpretation of dimension 4' to 219.81/61.11 orient following rules strictly. 219.81/61.11 219.81/61.11 Trs: { g(ok(X)) -> ok(g(X)) } 219.81/61.11 219.81/61.11 The induced complexity on above rules (modulo remaining rules) is 219.81/61.11 YES(?,O(n^2)) . These rules are moved into the corresponding weak 219.81/61.11 component(s). 219.81/61.11 219.81/61.11 Sub-proof: 219.81/61.11 ---------- 219.81/61.11 TcT has computed the following triangular matrix interpretation. 219.81/61.11 Note that the diagonal of the component-wise maxima of 219.81/61.11 interpretation-entries contains no more than 2 non-zero entries. 219.81/61.11 219.81/61.11 [1 0 1 0] [0] 219.81/61.11 [active](x1) = [0 0 0 0] x1 + [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 219.81/61.11 [1 1 1 0] [0] 219.81/61.11 [g](x1) = [0 1 0 0] x1 + [1] 219.81/61.11 [0 0 0 1] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 219.81/61.11 [1 1 1 0] [0] 219.81/61.11 [mark](x1) = [0 0 0 0] x1 + [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 219.81/61.11 [1 0 1 0] [0] 219.81/61.11 [h](x1) = [0 1 0 0] x1 + [0] 219.81/61.11 [0 0 0 1] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 219.81/61.11 [0] 219.81/61.11 [c] = [0] 219.81/61.11 [1] 219.81/61.11 [0] 219.81/61.11 219.81/61.11 [1] 219.81/61.11 [d] = [0] 219.81/61.11 [0] 219.81/61.11 [1] 219.81/61.11 219.81/61.11 [1 1 1 0] [0] 219.81/61.11 [proper](x1) = [0 1 0 0] x1 + [1] 219.81/61.11 [0 0 0 1] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 219.81/61.11 [1 0 1 0] [0] 219.81/61.11 [ok](x1) = [0 1 0 0] x1 + [1] 219.81/61.11 [0 0 0 1] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 219.81/61.11 [1 0 0 0] [0] 219.81/61.11 [top](x1) = [0 0 0 0] x1 + [0] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 219.81/61.11 The order satisfies the following ordering constraints: 219.81/61.11 219.81/61.11 [active(g(X))] = [1 1 1 1] [0] 219.81/61.11 [0 0 0 0] X + [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 >= [1 1 1 1] [0] 219.81/61.11 [0 0 0 0] X + [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 = [mark(h(X))] 219.81/61.11 219.81/61.11 [active(h(d()))] = [2] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 >= [2] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 = [mark(g(c()))] 219.81/61.11 219.81/61.11 [active(c())] = [1] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 >= [1] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 = [mark(d())] 219.81/61.11 219.81/61.11 [g(ok(X))] = [1 1 1 1] [1] 219.81/61.11 [0 1 0 0] X + [2] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 > [1 1 1 1] [0] 219.81/61.11 [0 1 0 0] X + [2] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 = [ok(g(X))] 219.81/61.11 219.81/61.11 [h(ok(X))] = [1 0 1 1] [0] 219.81/61.11 [0 1 0 0] X + [1] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 >= [1 0 1 1] [0] 219.81/61.11 [0 1 0 0] X + [1] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 = [ok(h(X))] 219.81/61.11 219.81/61.11 [proper(g(X))] = [1 2 1 1] [1] 219.81/61.11 [0 1 0 0] X + [2] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 >= [1 2 1 1] [1] 219.81/61.11 [0 1 0 0] X + [2] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 = [g(proper(X))] 219.81/61.11 219.81/61.11 [proper(h(X))] = [1 1 1 1] [0] 219.81/61.11 [0 1 0 0] X + [1] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 >= [1 1 1 1] [0] 219.81/61.11 [0 1 0 0] X + [1] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 [0 0 0 0] [0] 219.81/61.11 = [h(proper(X))] 219.81/61.11 219.81/61.11 [proper(c())] = [1] 219.81/61.11 [1] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 >= [1] 219.81/61.11 [1] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 = [ok(c())] 219.81/61.11 219.81/61.11 [proper(d())] = [1] 219.81/61.11 [1] 219.81/61.11 [1] 219.81/61.11 [0] 219.81/61.11 >= [1] 219.81/61.11 [1] 219.81/61.11 [1] 219.81/61.11 [0] 219.81/61.11 = [ok(d())] 219.81/61.11 219.81/61.11 [top(mark(X))] = [1 1 1 0] [0] 219.81/61.11 [0 0 0 0] X + [0] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 >= [1 1 1 0] [0] 219.81/61.11 [0 0 0 0] X + [0] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 = [top(proper(X))] 219.81/61.11 219.81/61.11 [top(ok(X))] = [1 0 1 0] [0] 219.81/61.11 [0 0 0 0] X + [0] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 >= [1 0 1 0] [0] 219.81/61.11 [0 0 0 0] X + [0] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 [0 0 0 0] [1] 219.81/61.11 = [top(active(X))] 219.81/61.11 219.81/61.11 219.81/61.11 We return to the main proof. 219.81/61.11 219.81/61.11 We are left with following problem, upon which TcT provides the 219.81/61.11 certificate YES(O(1),O(n^2)). 219.81/61.11 219.81/61.11 Strict Trs: { h(ok(X)) -> ok(h(X)) } 219.81/61.11 Weak Trs: 219.81/61.11 { active(g(X)) -> mark(h(X)) 219.81/61.11 , active(h(d())) -> mark(g(c())) 219.81/61.11 , active(c()) -> mark(d()) 219.81/61.11 , g(ok(X)) -> ok(g(X)) 219.81/61.11 , proper(g(X)) -> g(proper(X)) 219.81/61.11 , proper(h(X)) -> h(proper(X)) 219.81/61.11 , proper(c()) -> ok(c()) 219.81/61.11 , proper(d()) -> ok(d()) 219.81/61.11 , top(mark(X)) -> top(proper(X)) 219.81/61.11 , top(ok(X)) -> top(active(X)) } 219.81/61.11 Obligation: 219.81/61.11 derivational complexity 219.81/61.11 Answer: 219.81/61.11 YES(O(1),O(n^2)) 219.81/61.11 219.81/61.11 We use the processor 'matrix interpretation of dimension 3' to 219.81/61.11 orient following rules strictly. 219.81/61.11 219.81/61.11 Trs: { h(ok(X)) -> ok(h(X)) } 219.81/61.11 219.81/61.11 The induced complexity on above rules (modulo remaining rules) is 219.81/61.11 YES(?,O(n^2)) . These rules are moved into the corresponding weak 219.81/61.11 component(s). 219.81/61.11 219.81/61.11 Sub-proof: 219.81/61.11 ---------- 219.81/61.11 TcT has computed the following triangular matrix interpretation. 219.81/61.11 Note that the diagonal of the component-wise maxima of 219.81/61.11 interpretation-entries contains no more than 2 non-zero entries. 219.81/61.11 219.81/61.11 [1 1 0] [0] 219.81/61.11 [active](x1) = [0 0 0] x1 + [0] 219.81/61.11 [0 0 0] [0] 219.81/61.11 219.81/61.11 [1 2 2] [0] 219.81/61.11 [g](x1) = [0 1 2] x1 + [2] 219.81/61.11 [0 0 0] [0] 219.81/61.11 219.81/61.11 [1 2 0] [0] 219.81/61.11 [mark](x1) = [0 0 0] x1 + [0] 219.81/61.11 [0 0 0] [0] 219.81/61.11 219.81/61.11 [1 1 0] [0] 219.81/61.11 [h](x1) = [0 1 2] x1 + [1] 219.81/61.11 [0 0 0] [0] 219.81/61.11 219.81/61.11 [0] 219.81/61.11 [c] = [0] 219.81/61.11 [0] 219.81/61.11 219.81/61.11 [0] 219.81/61.11 [d] = [0] 219.81/61.11 [2] 219.81/61.11 219.81/61.11 [1 2 0] [0] 219.81/61.11 [proper](x1) = [0 1 2] x1 + [2] 219.81/61.11 [0 0 0] [0] 219.81/61.11 219.81/61.11 [1 1 0] [0] 219.81/61.11 [ok](x1) = [0 1 2] x1 + [2] 219.81/61.11 [0 0 0] [0] 219.81/61.11 219.81/61.11 [1 0 0] [0] 219.81/61.11 [top](x1) = [0 0 0] x1 + [0] 219.81/61.11 [0 0 0] [0] 219.81/61.11 219.81/61.11 The order satisfies the following ordering constraints: 219.81/61.11 219.81/61.11 [active(g(X))] = [1 3 4] [2] 219.81/61.11 [0 0 0] X + [0] 219.81/61.11 [0 0 0] [0] 219.81/61.11 >= [1 3 4] [2] 219.81/61.11 [0 0 0] X + [0] 219.81/61.11 [0 0 0] [0] 219.81/61.11 = [mark(h(X))] 219.81/61.11 219.81/61.11 [active(h(d()))] = [5] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 > [4] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 = [mark(g(c()))] 219.81/61.11 219.81/61.11 [active(c())] = [0] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 >= [0] 219.81/61.11 [0] 219.81/61.11 [0] 219.81/61.11 = [mark(d())] 219.81/61.11 219.81/61.11 [g(ok(X))] = [1 3 4] [4] 219.81/61.11 [0 1 2] X + [4] 219.81/61.11 [0 0 0] [0] 219.81/61.11 > [1 3 4] [2] 219.81/61.11 [0 1 2] X + [4] 219.81/61.11 [0 0 0] [0] 219.81/61.11 = [ok(g(X))] 219.81/61.11 219.81/61.11 [h(ok(X))] = [1 2 2] [2] 219.81/61.11 [0 1 2] X + [3] 219.81/61.11 [0 0 0] [0] 219.81/61.11 > [1 2 2] [1] 219.81/61.11 [0 1 2] X + [3] 219.81/61.11 [0 0 0] [0] 219.81/61.11 = [ok(h(X))] 219.81/61.11 219.81/61.11 [proper(g(X))] = [1 4 6] [4] 219.81/61.11 [0 1 2] X + [4] 219.81/61.11 [0 0 0] [0] 219.81/61.11 >= [1 4 4] [4] 219.81/61.11 [0 1 2] X + [4] 219.81/61.11 [0 0 0] [0] 219.81/61.11 = [g(proper(X))] 219.81/61.11 219.81/61.11 [proper(h(X))] = [1 3 4] [2] 219.81/61.11 [0 1 2] X + [3] 219.81/61.11 [0 0 0] [0] 219.81/61.11 >= [1 3 2] [2] 219.81/61.11 [0 1 2] X + [3] 219.81/61.11 [0 0 0] [0] 219.81/61.11 = [h(proper(X))] 219.81/61.11 219.81/61.11 [proper(c())] = [0] 219.81/61.11 [2] 219.81/61.11 [0] 219.81/61.11 >= [0] 219.81/61.11 [2] 219.81/61.11 [0] 219.81/61.11 = [ok(c())] 219.81/61.11 219.81/61.11 [proper(d())] = [0] 219.81/61.11 [6] 219.81/61.11 [0] 219.81/61.11 >= [0] 219.81/61.11 [6] 219.81/61.11 [0] 219.81/61.11 = [ok(d())] 219.81/61.11 219.81/61.11 [top(mark(X))] = [1 2 0] [0] 219.81/61.11 [0 0 0] X + [0] 219.81/61.11 [0 0 0] [0] 219.81/61.11 >= [1 2 0] [0] 219.81/61.11 [0 0 0] X + [0] 219.81/61.11 [0 0 0] [0] 219.81/61.11 = [top(proper(X))] 219.81/61.11 219.81/61.11 [top(ok(X))] = [1 1 0] [0] 219.81/61.11 [0 0 0] X + [0] 219.81/61.11 [0 0 0] [0] 219.81/61.11 >= [1 1 0] [0] 219.81/61.11 [0 0 0] X + [0] 219.81/61.11 [0 0 0] [0] 219.81/61.11 = [top(active(X))] 219.81/61.11 219.81/61.11 219.81/61.11 We return to the main proof. 219.81/61.11 219.81/61.11 We are left with following problem, upon which TcT provides the 219.81/61.11 certificate YES(O(1),O(1)). 219.81/61.11 219.81/61.11 Weak Trs: 219.81/61.11 { active(g(X)) -> mark(h(X)) 219.81/61.11 , active(h(d())) -> mark(g(c())) 219.81/61.11 , active(c()) -> mark(d()) 219.81/61.11 , g(ok(X)) -> ok(g(X)) 219.81/61.11 , h(ok(X)) -> ok(h(X)) 219.81/61.11 , proper(g(X)) -> g(proper(X)) 219.81/61.11 , proper(h(X)) -> h(proper(X)) 219.81/61.11 , proper(c()) -> ok(c()) 219.81/61.11 , proper(d()) -> ok(d()) 219.81/61.11 , top(mark(X)) -> top(proper(X)) 219.81/61.11 , top(ok(X)) -> top(active(X)) } 219.81/61.11 Obligation: 219.81/61.11 derivational complexity 219.81/61.11 Answer: 219.81/61.11 YES(O(1),O(1)) 219.81/61.11 219.81/61.11 Empty rules are trivially bounded 219.81/61.11 219.81/61.11 Hurray, we answered YES(O(1),O(n^2)) 219.81/61.12 EOF