YES(O(1),O(n^2)) 163.20/60.06 YES(O(1),O(n^2)) 163.20/60.06 163.20/60.06 We are left with following problem, upon which TcT provides the 163.20/60.06 certificate YES(O(1),O(n^2)). 163.20/60.06 163.20/60.06 Strict Trs: 163.20/60.06 { implies(x, or(y, z)) -> or(y, implies(x, z)) 163.20/60.06 , implies(not(x), y) -> or(x, y) 163.20/60.06 , implies(not(x), or(y, z)) -> implies(y, or(x, z)) } 163.20/60.06 Obligation: 163.20/60.06 derivational complexity 163.20/60.06 Answer: 163.20/60.06 YES(O(1),O(n^2)) 163.20/60.06 163.20/60.06 We use the processor 'matrix interpretation of dimension 1' to 163.20/60.06 orient following rules strictly. 163.20/60.06 163.20/60.06 Trs: 163.20/60.06 { implies(not(x), y) -> or(x, y) 163.20/60.06 , implies(not(x), or(y, z)) -> implies(y, or(x, z)) } 163.20/60.06 163.20/60.06 The induced complexity on above rules (modulo remaining rules) is 163.20/60.06 YES(?,O(n^1)) . These rules are moved into the corresponding weak 163.20/60.06 component(s). 163.20/60.06 163.20/60.06 Sub-proof: 163.20/60.06 ---------- 163.20/60.06 TcT has computed the following triangular matrix interpretation. 163.20/60.06 163.20/60.06 [implies](x1, x2) = [1] x1 + [1] x2 + [0] 163.20/60.06 163.20/60.06 [not](x1) = [1] x1 + [1] 163.20/60.06 163.20/60.06 [or](x1, x2) = [1] x1 + [1] x2 + [0] 163.20/60.06 163.20/60.06 The order satisfies the following ordering constraints: 163.20/60.06 163.20/60.06 [implies(x, or(y, z))] = [1] x + [1] y + [1] z + [0] 163.20/60.06 >= [1] x + [1] y + [1] z + [0] 163.20/60.06 = [or(y, implies(x, z))] 163.20/60.06 163.20/60.06 [implies(not(x), y)] = [1] x + [1] y + [1] 163.20/60.06 > [1] x + [1] y + [0] 163.20/60.06 = [or(x, y)] 163.20/60.06 163.20/60.06 [implies(not(x), or(y, z))] = [1] x + [1] y + [1] z + [1] 163.20/60.06 > [1] x + [1] y + [1] z + [0] 163.20/60.06 = [implies(y, or(x, z))] 163.20/60.06 163.20/60.06 163.20/60.06 We return to the main proof. 163.20/60.06 163.20/60.06 We are left with following problem, upon which TcT provides the 163.20/60.06 certificate YES(O(1),O(n^2)). 163.20/60.06 163.20/60.06 Strict Trs: { implies(x, or(y, z)) -> or(y, implies(x, z)) } 163.20/60.06 Weak Trs: 163.20/60.06 { implies(not(x), y) -> or(x, y) 163.20/60.06 , implies(not(x), or(y, z)) -> implies(y, or(x, z)) } 163.20/60.06 Obligation: 163.20/60.06 derivational complexity 163.20/60.06 Answer: 163.20/60.06 YES(O(1),O(n^2)) 163.20/60.06 163.20/60.06 We use the processor 'matrix interpretation of dimension 2' to 163.20/60.06 orient following rules strictly. 163.20/60.06 163.20/60.06 Trs: { implies(x, or(y, z)) -> or(y, implies(x, z)) } 163.20/60.06 163.20/60.06 The induced complexity on above rules (modulo remaining rules) is 163.20/60.06 YES(?,O(n^2)) . These rules are moved into the corresponding weak 163.20/60.06 component(s). 163.20/60.06 163.20/60.06 Sub-proof: 163.20/60.06 ---------- 163.20/60.06 TcT has computed the following triangular matrix interpretation. 163.20/60.06 163.20/60.06 [implies](x1, x2) = [1 2] x1 + [1 1] x2 + [0] 163.20/60.06 [0 0] [0 1] [1] 163.20/60.06 163.20/60.06 [not](x1) = [1 2] x1 + [0] 163.20/60.06 [0 1] [2] 163.20/60.06 163.20/60.06 [or](x1, x2) = [1 2] x1 + [1 0] x2 + [0] 163.20/60.06 [0 0] [0 1] [1] 163.20/60.06 163.20/60.06 The order satisfies the following ordering constraints: 163.20/60.06 163.20/60.06 [implies(x, or(y, z))] = [1 2] x + [1 2] y + [1 1] z + [1] 163.20/60.06 [0 0] [0 0] [0 1] [2] 163.20/60.06 > [1 2] x + [1 2] y + [1 1] z + [0] 163.20/60.06 [0 0] [0 0] [0 1] [2] 163.20/60.06 = [or(y, implies(x, z))] 163.20/60.06 163.20/60.06 [implies(not(x), y)] = [1 4] x + [1 1] y + [4] 163.20/60.06 [0 0] [0 1] [1] 163.20/60.06 > [1 2] x + [1 0] y + [0] 163.20/60.06 [0 0] [0 1] [1] 163.20/60.06 = [or(x, y)] 163.20/60.06 163.20/60.06 [implies(not(x), or(y, z))] = [1 4] x + [1 2] y + [1 1] z + [5] 163.20/60.06 [0 0] [0 0] [0 1] [2] 163.20/60.06 > [1 2] x + [1 2] y + [1 1] z + [1] 163.20/60.06 [0 0] [0 0] [0 1] [2] 163.20/60.06 = [implies(y, or(x, z))] 163.20/60.06 163.20/60.06 163.20/60.06 We return to the main proof. 163.20/60.06 163.20/60.06 We are left with following problem, upon which TcT provides the 163.20/60.06 certificate YES(O(1),O(1)). 163.20/60.06 163.20/60.06 Weak Trs: 163.20/60.06 { implies(x, or(y, z)) -> or(y, implies(x, z)) 163.20/60.06 , implies(not(x), y) -> or(x, y) 163.20/60.06 , implies(not(x), or(y, z)) -> implies(y, or(x, z)) } 163.20/60.06 Obligation: 163.20/60.06 derivational complexity 163.20/60.06 Answer: 163.20/60.06 YES(O(1),O(1)) 163.20/60.06 163.20/60.06 Empty rules are trivially bounded 163.20/60.06 163.20/60.06 Hurray, we answered YES(O(1),O(n^2)) 163.20/60.07 EOF