YES(O(1),O(n^1)) 35.65/13.73 YES(O(1),O(n^1)) 35.65/13.73 35.65/13.73 We are left with following problem, upon which TcT provides the 35.65/13.73 certificate YES(O(1),O(n^1)). 35.65/13.73 35.65/13.73 Strict Trs: 35.65/13.73 { -(x, 0()) -> x 35.65/13.73 , -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0()) 35.65/13.73 , -(0(), y) -> 0() 35.65/13.73 , p(0()) -> 0() 35.65/13.73 , p(s(x)) -> x } 35.65/13.73 Obligation: 35.65/13.73 innermost runtime complexity 35.65/13.73 Answer: 35.65/13.73 YES(O(1),O(n^1)) 35.65/13.73 35.65/13.73 The weightgap principle applies (using the following nonconstant 35.65/13.73 growth matrix-interpretation) 35.65/13.73 35.65/13.73 The following argument positions are usable: 35.65/13.73 Uargs(-) = {2}, Uargs(s) = {1}, Uargs(if) = {2} 35.65/13.73 35.65/13.73 TcT has computed the following matrix interpretation satisfying 35.65/13.73 not(EDA) and not(IDA(1)). 35.65/13.73 35.65/13.73 [-](x1, x2) = [1] x1 + [1] x2 + [0] 35.65/13.73 35.65/13.73 [0] = [0] 35.65/13.73 35.65/13.73 [s](x1) = [1] x1 + [0] 35.65/13.73 35.65/13.73 [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1] 35.65/13.73 35.65/13.73 [greater](x1, x2) = [1] 35.65/13.73 35.65/13.73 [p](x1) = [1] x1 + [4] 35.65/13.73 35.65/13.73 The order satisfies the following ordering constraints: 35.65/13.73 35.65/13.73 [-(x, 0())] = [1] x + [0] 35.65/13.73 >= [1] x + [0] 35.65/13.73 = [x] 35.65/13.73 35.65/13.73 [-(x, s(y))] = [1] y + [1] x + [0] 35.65/13.73 ? [1] y + [1] x + [6] 35.65/13.73 = [if(greater(x, s(y)), s(-(x, p(s(y)))), 0())] 35.65/13.73 35.65/13.73 [-(0(), y)] = [1] y + [0] 35.65/13.73 >= [0] 35.65/13.73 = [0()] 35.65/13.73 35.65/13.73 [p(0())] = [4] 35.65/13.73 > [0] 35.65/13.73 = [0()] 35.65/13.73 35.65/13.73 [p(s(x))] = [1] x + [4] 35.65/13.73 > [1] x + [0] 35.65/13.73 = [x] 35.65/13.73 35.65/13.73 35.65/13.73 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 35.65/13.73 35.65/13.73 We are left with following problem, upon which TcT provides the 35.65/13.73 certificate YES(O(1),O(n^1)). 35.65/13.73 35.65/13.73 Strict Trs: 35.65/13.73 { -(x, 0()) -> x 35.65/13.73 , -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0()) 35.65/13.73 , -(0(), y) -> 0() } 35.65/13.73 Weak Trs: 35.65/13.73 { p(0()) -> 0() 35.65/13.73 , p(s(x)) -> x } 35.65/13.73 Obligation: 35.65/13.73 innermost runtime complexity 35.65/13.73 Answer: 35.65/13.73 YES(O(1),O(n^1)) 35.65/13.73 35.65/13.73 The weightgap principle applies (using the following nonconstant 35.65/13.73 growth matrix-interpretation) 35.65/13.73 35.65/13.73 The following argument positions are usable: 35.65/13.73 Uargs(-) = {2}, Uargs(s) = {1}, Uargs(if) = {2} 35.65/13.73 35.65/13.73 TcT has computed the following matrix interpretation satisfying 35.65/13.73 not(EDA) and not(IDA(1)). 35.65/13.73 35.65/13.73 [-](x1, x2) = [1] x1 + [1] x2 + [1] 35.65/13.73 35.65/13.73 [0] = [0] 35.65/13.73 35.65/13.73 [s](x1) = [1] x1 + [0] 35.65/13.73 35.65/13.73 [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1] 35.65/13.73 35.65/13.73 [greater](x1, x2) = [0] 35.65/13.73 35.65/13.73 [p](x1) = [1] x1 + [0] 35.65/13.73 35.65/13.73 The order satisfies the following ordering constraints: 35.65/13.73 35.65/13.73 [-(x, 0())] = [1] x + [1] 35.65/13.73 > [1] x + [0] 35.65/13.73 = [x] 35.65/13.73 35.65/13.73 [-(x, s(y))] = [1] y + [1] x + [1] 35.65/13.73 ? [1] y + [1] x + [2] 35.65/13.73 = [if(greater(x, s(y)), s(-(x, p(s(y)))), 0())] 35.65/13.73 35.65/13.73 [-(0(), y)] = [1] y + [1] 35.65/13.73 > [0] 35.65/13.73 = [0()] 35.65/13.73 35.65/13.73 [p(0())] = [0] 35.65/13.73 >= [0] 35.65/13.73 = [0()] 35.65/13.73 35.65/13.73 [p(s(x))] = [1] x + [0] 35.65/13.73 >= [1] x + [0] 35.65/13.73 = [x] 35.65/13.73 35.65/13.73 35.65/13.73 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 35.65/13.73 35.65/13.73 We are left with following problem, upon which TcT provides the 35.65/13.73 certificate YES(O(1),O(n^1)). 35.65/13.73 35.65/13.73 Strict Trs: 35.65/13.73 { -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0()) } 35.65/13.73 Weak Trs: 35.65/13.73 { -(x, 0()) -> x 35.65/13.73 , -(0(), y) -> 0() 35.65/13.73 , p(0()) -> 0() 35.65/13.73 , p(s(x)) -> x } 35.65/13.73 Obligation: 35.65/13.73 innermost runtime complexity 35.65/13.73 Answer: 35.65/13.73 YES(O(1),O(n^1)) 35.65/13.73 35.65/13.73 We use the processor 'matrix interpretation of dimension 3' to 35.65/13.73 orient following rules strictly. 35.65/13.73 35.65/13.73 Trs: { -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0()) } 35.65/13.73 35.65/13.73 The induced complexity on above rules (modulo remaining rules) is 35.65/13.73 YES(?,O(n^1)) . These rules are moved into the corresponding weak 35.65/13.73 component(s). 35.65/13.73 35.65/13.73 Sub-proof: 35.65/13.73 ---------- 35.65/13.73 The following argument positions are usable: 35.65/13.73 Uargs(-) = {2}, Uargs(s) = {1}, Uargs(if) = {2} 35.65/13.73 35.65/13.73 TcT has computed the following constructor-based matrix 35.65/13.73 interpretation satisfying not(EDA) and not(IDA(1)). 35.65/13.73 35.65/13.73 [4 0 0] [4 0 0] [0] 35.65/13.73 [-](x1, x2) = [4 1 4] x1 + [0 0 0] x2 + [0] 35.65/13.73 [0 0 4] [0 4 0] [4] 35.65/13.73 35.65/13.73 [0] 35.65/13.73 [0] = [0] 35.65/13.73 [0] 35.65/13.73 35.65/13.73 [1 0 0] [2] 35.65/13.73 [s](x1) = [0 1 2] x1 + [0] 35.65/13.73 [1 0 0] [0] 35.65/13.73 35.65/13.73 [1 0 0] [0] 35.65/13.73 [if](x1, x2, x3) = [0 0 0] x2 + [0] 35.65/13.73 [0 0 0] [3] 35.65/13.73 35.65/13.73 [7] 35.65/13.73 [greater](x1, x2) = [0] 35.65/13.73 [0] 35.65/13.73 35.65/13.73 [0 0 1] [0] 35.65/13.73 [p](x1) = [0 4 0] x1 + [4] 35.65/13.73 [0 1 0] [0] 35.65/13.73 35.65/13.73 The order satisfies the following ordering constraints: 35.65/13.73 35.65/13.73 [-(x, 0())] = [4 0 0] [0] 35.65/13.73 [4 1 4] x + [0] 35.65/13.73 [0 0 4] [4] 35.65/13.73 >= [1 0 0] [0] 35.65/13.73 [0 1 0] x + [0] 35.65/13.73 [0 0 1] [0] 35.65/13.73 = [x] 35.65/13.73 35.65/13.73 [-(x, s(y))] = [4 0 0] [4 0 0] [8] 35.65/13.73 [0 0 0] y + [4 1 4] x + [0] 35.65/13.73 [0 4 8] [0 0 4] [4] 35.65/13.73 > [4 0 0] [4 0 0] [2] 35.65/13.73 [0 0 0] y + [0 0 0] x + [0] 35.65/13.73 [0 0 0] [0 0 0] [3] 35.65/13.73 = [if(greater(x, s(y)), s(-(x, p(s(y)))), 0())] 35.65/13.73 35.65/13.73 [-(0(), y)] = [4 0 0] [0] 35.65/13.73 [0 0 0] y + [0] 35.65/13.73 [0 4 0] [4] 35.65/13.73 >= [0] 35.65/13.73 [0] 35.65/13.73 [0] 35.65/13.73 = [0()] 35.65/13.73 35.65/13.73 [p(0())] = [0] 35.65/13.73 [4] 35.65/13.73 [0] 35.65/13.73 >= [0] 35.65/13.73 [0] 35.65/13.73 [0] 35.65/13.73 = [0()] 35.65/13.73 35.65/13.73 [p(s(x))] = [1 0 0] [0] 35.65/13.73 [0 4 8] x + [4] 35.65/13.73 [0 1 2] [0] 35.65/13.73 >= [1 0 0] [0] 35.65/13.73 [0 1 0] x + [0] 35.65/13.73 [0 0 1] [0] 35.65/13.73 = [x] 35.65/13.73 35.65/13.73 35.65/13.73 We return to the main proof. 35.65/13.73 35.65/13.73 We are left with following problem, upon which TcT provides the 35.65/13.73 certificate YES(O(1),O(1)). 35.65/13.73 35.65/13.73 Weak Trs: 35.65/13.73 { -(x, 0()) -> x 35.65/13.73 , -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0()) 35.65/13.73 , -(0(), y) -> 0() 35.65/13.73 , p(0()) -> 0() 35.65/13.73 , p(s(x)) -> x } 35.65/13.73 Obligation: 35.65/13.73 innermost runtime complexity 35.65/13.73 Answer: 35.65/13.73 YES(O(1),O(1)) 35.65/13.73 35.65/13.73 Empty rules are trivially bounded 35.65/13.73 35.65/13.73 Hurray, we answered YES(O(1),O(n^1)) 35.65/13.77 EOF