YES(O(1),O(n^2)) 168.71/60.06 YES(O(1),O(n^2)) 168.71/60.06 168.71/60.06 We are left with following problem, upon which TcT provides the 168.71/60.06 certificate YES(O(1),O(n^2)). 168.71/60.06 168.71/60.06 Strict Trs: 168.71/60.06 { w(r(x)) -> r(w(x)) 168.71/60.06 , b(w(x)) -> w(b(x)) 168.71/60.06 , b(r(x)) -> r(b(x)) } 168.71/60.06 Obligation: 168.71/60.06 derivational complexity 168.71/60.06 Answer: 168.71/60.06 YES(O(1),O(n^2)) 168.71/60.06 168.71/60.06 We use the processor 'matrix interpretation of dimension 2' to 168.71/60.06 orient following rules strictly. 168.71/60.06 168.71/60.06 Trs: { b(r(x)) -> r(b(x)) } 168.71/60.06 168.71/60.06 The induced complexity on above rules (modulo remaining rules) is 168.71/60.06 YES(?,O(n^2)) . These rules are moved into the corresponding weak 168.71/60.06 component(s). 168.71/60.06 168.71/60.06 Sub-proof: 168.71/60.06 ---------- 168.71/60.06 TcT has computed the following triangular matrix interpretation. 168.71/60.06 168.71/60.06 [w](x1) = [1 0] x1 + [0] 168.71/60.06 [0 1] [0] 168.71/60.06 168.71/60.06 [r](x1) = [1 0] x1 + [0] 168.71/60.06 [0 1] [1] 168.71/60.06 168.71/60.06 [b](x1) = [1 1] x1 + [0] 168.71/60.06 [0 1] [0] 168.71/60.06 168.71/60.06 The order satisfies the following ordering constraints: 168.71/60.06 168.71/60.06 [w(r(x))] = [1 0] x + [0] 168.71/60.06 [0 1] [1] 168.71/60.06 >= [1 0] x + [0] 168.71/60.06 [0 1] [1] 168.71/60.06 = [r(w(x))] 168.71/60.06 168.71/60.06 [b(w(x))] = [1 1] x + [0] 168.71/60.06 [0 1] [0] 168.71/60.06 >= [1 1] x + [0] 168.71/60.06 [0 1] [0] 168.71/60.06 = [w(b(x))] 168.71/60.06 168.71/60.06 [b(r(x))] = [1 1] x + [1] 168.71/60.06 [0 1] [1] 168.71/60.06 > [1 1] x + [0] 168.71/60.06 [0 1] [1] 168.71/60.06 = [r(b(x))] 168.71/60.06 168.71/60.06 168.71/60.06 We return to the main proof. 168.71/60.06 168.71/60.06 We are left with following problem, upon which TcT provides the 168.71/60.06 certificate YES(O(1),O(n^2)). 168.71/60.06 168.71/60.06 Strict Trs: 168.71/60.06 { w(r(x)) -> r(w(x)) 168.71/60.06 , b(w(x)) -> w(b(x)) } 168.71/60.06 Weak Trs: { b(r(x)) -> r(b(x)) } 168.71/60.06 Obligation: 168.71/60.06 derivational complexity 168.71/60.06 Answer: 168.71/60.06 YES(O(1),O(n^2)) 168.71/60.06 168.71/60.06 We use the processor 'matrix interpretation of dimension 2' to 168.71/60.06 orient following rules strictly. 168.71/60.06 168.71/60.06 Trs: { b(w(x)) -> w(b(x)) } 168.71/60.06 168.71/60.06 The induced complexity on above rules (modulo remaining rules) is 168.71/60.06 YES(?,O(n^2)) . These rules are moved into the corresponding weak 168.71/60.06 component(s). 168.71/60.06 168.71/60.06 Sub-proof: 168.71/60.06 ---------- 168.71/60.06 TcT has computed the following triangular matrix interpretation. 168.71/60.06 168.71/60.06 [w](x1) = [1 0] x1 + [0] 168.71/60.06 [0 1] [12] 168.71/60.06 168.71/60.06 [r](x1) = [1 0] x1 + [0] 168.71/60.06 [0 1] [20] 168.71/60.06 168.71/60.06 [b](x1) = [1 1] x1 + [20] 168.71/60.06 [0 1] [0] 168.71/60.06 168.71/60.06 The order satisfies the following ordering constraints: 168.71/60.06 168.71/60.06 [w(r(x))] = [1 0] x + [0] 168.71/60.06 [0 1] [32] 168.71/60.06 >= [1 0] x + [0] 168.71/60.06 [0 1] [32] 168.71/60.06 = [r(w(x))] 168.71/60.06 168.71/60.06 [b(w(x))] = [1 1] x + [32] 168.71/60.06 [0 1] [12] 168.71/60.06 > [1 1] x + [20] 168.71/60.06 [0 1] [12] 168.71/60.06 = [w(b(x))] 168.71/60.06 168.71/60.06 [b(r(x))] = [1 1] x + [40] 168.71/60.06 [0 1] [20] 168.71/60.06 > [1 1] x + [20] 168.71/60.06 [0 1] [20] 168.71/60.06 = [r(b(x))] 168.71/60.06 168.71/60.06 168.71/60.06 We return to the main proof. 168.71/60.06 168.71/60.06 We are left with following problem, upon which TcT provides the 168.71/60.06 certificate YES(O(1),O(n^2)). 168.71/60.06 168.71/60.06 Strict Trs: { w(r(x)) -> r(w(x)) } 168.71/60.06 Weak Trs: 168.71/60.06 { b(w(x)) -> w(b(x)) 168.71/60.06 , b(r(x)) -> r(b(x)) } 168.71/60.06 Obligation: 168.71/60.06 derivational complexity 168.71/60.06 Answer: 168.71/60.06 YES(O(1),O(n^2)) 168.71/60.06 168.71/60.06 We use the processor 'matrix interpretation of dimension 2' to 168.71/60.06 orient following rules strictly. 168.71/60.06 168.71/60.06 Trs: { w(r(x)) -> r(w(x)) } 168.71/60.06 168.71/60.06 The induced complexity on above rules (modulo remaining rules) is 168.71/60.06 YES(?,O(n^2)) . These rules are moved into the corresponding weak 168.71/60.06 component(s). 168.71/60.06 168.71/60.06 Sub-proof: 168.71/60.06 ---------- 168.71/60.06 TcT has computed the following triangular matrix interpretation. 168.71/60.06 168.71/60.06 [w](x1) = [1 1] x1 + [0] 168.71/60.06 [0 1] [0] 168.71/60.06 168.71/60.06 [r](x1) = [1 0] x1 + [0] 168.71/60.06 [0 1] [1] 168.71/60.06 168.71/60.06 [b](x1) = [1 0] x1 + [0] 168.71/60.06 [0 1] [0] 168.71/60.06 168.71/60.06 The order satisfies the following ordering constraints: 168.71/60.06 168.71/60.06 [w(r(x))] = [1 1] x + [1] 168.71/60.06 [0 1] [1] 168.71/60.06 > [1 1] x + [0] 168.71/60.06 [0 1] [1] 168.71/60.06 = [r(w(x))] 168.71/60.06 168.71/60.06 [b(w(x))] = [1 1] x + [0] 168.71/60.06 [0 1] [0] 168.71/60.06 >= [1 1] x + [0] 168.71/60.06 [0 1] [0] 168.71/60.06 = [w(b(x))] 168.71/60.06 168.71/60.06 [b(r(x))] = [1 0] x + [0] 168.71/60.06 [0 1] [1] 168.71/60.06 >= [1 0] x + [0] 168.71/60.06 [0 1] [1] 168.71/60.06 = [r(b(x))] 168.71/60.06 168.71/60.06 168.71/60.06 We return to the main proof. 168.71/60.06 168.71/60.06 We are left with following problem, upon which TcT provides the 168.71/60.06 certificate YES(O(1),O(1)). 168.71/60.06 168.71/60.06 Weak Trs: 168.71/60.06 { w(r(x)) -> r(w(x)) 168.71/60.06 , b(w(x)) -> w(b(x)) 168.71/60.06 , b(r(x)) -> r(b(x)) } 168.71/60.06 Obligation: 168.71/60.06 derivational complexity 168.71/60.06 Answer: 168.71/60.06 YES(O(1),O(1)) 168.71/60.06 168.71/60.06 Empty rules are trivially bounded 168.71/60.06 168.71/60.06 Hurray, we answered YES(O(1),O(n^2)) 168.71/60.07 EOF