YES(O(1),O(n^1)) 0.00/0.72 YES(O(1),O(n^1)) 0.00/0.72 0.00/0.72 We are left with following problem, upon which TcT provides the 0.00/0.72 certificate YES(O(1),O(n^1)). 0.00/0.72 0.00/0.72 Strict Trs: 0.00/0.72 { D(t()) -> 1() 0.00/0.72 , D(constant()) -> 0() 0.00/0.72 , D(+(x, y)) -> +(D(x), D(y)) 0.00/0.72 , D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) 0.00/0.72 , D(-(x, y)) -> -(D(x), D(y)) } 0.00/0.72 Obligation: 0.00/0.72 runtime complexity 0.00/0.72 Answer: 0.00/0.72 YES(O(1),O(n^1)) 0.00/0.72 0.00/0.72 We add the following weak dependency pairs: 0.00/0.72 0.00/0.72 Strict DPs: 0.00/0.72 { D^#(t()) -> c_1() 0.00/0.72 , D^#(constant()) -> c_2() 0.00/0.72 , D^#(+(x, y)) -> c_3(D^#(x), D^#(y)) 0.00/0.72 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y)) 0.00/0.72 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) } 0.00/0.72 0.00/0.72 and mark the set of starting terms. 0.00/0.72 0.00/0.72 We are left with following problem, upon which TcT provides the 0.00/0.72 certificate YES(O(1),O(n^1)). 0.00/0.72 0.00/0.72 Strict DPs: 0.00/0.72 { D^#(t()) -> c_1() 0.00/0.72 , D^#(constant()) -> c_2() 0.00/0.72 , D^#(+(x, y)) -> c_3(D^#(x), D^#(y)) 0.00/0.72 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y)) 0.00/0.72 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) } 0.00/0.72 Strict Trs: 0.00/0.72 { D(t()) -> 1() 0.00/0.72 , D(constant()) -> 0() 0.00/0.72 , D(+(x, y)) -> +(D(x), D(y)) 0.00/0.72 , D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) 0.00/0.72 , D(-(x, y)) -> -(D(x), D(y)) } 0.00/0.72 Obligation: 0.00/0.72 runtime complexity 0.00/0.72 Answer: 0.00/0.72 YES(O(1),O(n^1)) 0.00/0.72 0.00/0.72 No rule is usable, rules are removed from the input problem. 0.00/0.72 0.00/0.72 We are left with following problem, upon which TcT provides the 0.00/0.72 certificate YES(O(1),O(n^1)). 0.00/0.72 0.00/0.72 Strict DPs: 0.00/0.72 { D^#(t()) -> c_1() 0.00/0.72 , D^#(constant()) -> c_2() 0.00/0.72 , D^#(+(x, y)) -> c_3(D^#(x), D^#(y)) 0.00/0.72 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y)) 0.00/0.72 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) } 0.00/0.72 Obligation: 0.00/0.72 runtime complexity 0.00/0.72 Answer: 0.00/0.72 YES(O(1),O(n^1)) 0.00/0.72 0.00/0.72 The weightgap principle applies (using the following constant 0.00/0.72 growth matrix-interpretation) 0.00/0.72 0.00/0.72 The following argument positions are usable: 0.00/0.72 Uargs(c_3) = {1, 2}, Uargs(c_4) = {2, 4}, Uargs(c_5) = {1, 2} 0.00/0.72 0.00/0.72 TcT has computed the following constructor-restricted matrix 0.00/0.72 interpretation. 0.00/0.72 0.00/0.72 [t] = [0] 0.00/0.72 [0] 0.00/0.72 0.00/0.72 [constant] = [0] 0.00/0.72 [0] 0.00/0.72 0.00/0.72 [+](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 0.00/0.72 [0 0] [0 0] [0] 0.00/0.72 0.00/0.72 [*](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 0.00/0.72 [0 0] [0 0] [0] 0.00/0.72 0.00/0.72 [-](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 0.00/0.72 [0 0] [0 0] [0] 0.00/0.72 0.00/0.72 [D^#](x1) = [1] 0.00/0.72 [0] 0.00/0.72 0.00/0.72 [c_1] = [0] 0.00/0.72 [0] 0.00/0.72 0.00/0.72 [c_2] = [0] 0.00/0.72 [0] 0.00/0.72 0.00/0.72 [c_3](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 0.00/0.72 [0 1] [0 1] [0] 0.00/0.72 0.00/0.72 [c_4](x1, x2, x3, x4) = [0 0] x1 + [1 0] x2 + [0 0] x3 + [1 0.00/0.72 0] x4 + [2] 0.00/0.72 [2 0] [0 1] [2 0] [0 0.00/0.72 1] [0] 0.00/0.72 0.00/0.72 [c_5](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 0.00/0.72 [0 1] [0 1] [2] 0.00/0.72 0.00/0.72 The order satisfies the following ordering constraints: 0.00/0.72 0.00/0.72 [D^#(t())] = [1] 0.00/0.72 [0] 0.00/0.72 > [0] 0.00/0.72 [0] 0.00/0.72 = [c_1()] 0.00/0.72 0.00/0.72 [D^#(constant())] = [1] 0.00/0.72 [0] 0.00/0.72 > [0] 0.00/0.72 [0] 0.00/0.72 = [c_2()] 0.00/0.72 0.00/0.72 [D^#(+(x, y))] = [1] 0.00/0.72 [0] 0.00/0.72 ? [4] 0.00/0.72 [0] 0.00/0.72 = [c_3(D^#(x), D^#(y))] 0.00/0.72 0.00/0.72 [D^#(*(x, y))] = [1] 0.00/0.72 [0] 0.00/0.72 ? [0 0] x + [0 0] y + [4] 0.00/0.72 [2 0] [2 0] [0] 0.00/0.72 = [c_4(y, D^#(x), x, D^#(y))] 0.00/0.72 0.00/0.72 [D^#(-(x, y))] = [1] 0.00/0.72 [0] 0.00/0.72 ? [4] 0.00/0.72 [2] 0.00/0.72 = [c_5(D^#(x), D^#(y))] 0.00/0.72 0.00/0.72 0.00/0.72 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 0.00/0.72 0.00/0.72 We are left with following problem, upon which TcT provides the 0.00/0.72 certificate YES(O(1),O(n^1)). 0.00/0.72 0.00/0.72 Strict DPs: 0.00/0.72 { D^#(+(x, y)) -> c_3(D^#(x), D^#(y)) 0.00/0.72 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y)) 0.00/0.72 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) } 0.00/0.72 Weak DPs: 0.00/0.72 { D^#(t()) -> c_1() 0.00/0.72 , D^#(constant()) -> c_2() } 0.00/0.72 Obligation: 0.00/0.72 runtime complexity 0.00/0.72 Answer: 0.00/0.72 YES(O(1),O(n^1)) 0.00/0.72 0.00/0.72 The following weak DPs constitute a sub-graph of the DG that is 0.00/0.72 closed under successors. The DPs are removed. 0.00/0.72 0.00/0.72 { D^#(t()) -> c_1() 0.00/0.72 , D^#(constant()) -> c_2() } 0.00/0.72 0.00/0.72 We are left with following problem, upon which TcT provides the 0.00/0.72 certificate YES(O(1),O(n^1)). 0.00/0.72 0.00/0.72 Strict DPs: 0.00/0.72 { D^#(+(x, y)) -> c_3(D^#(x), D^#(y)) 0.00/0.72 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y)) 0.00/0.72 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) } 0.00/0.72 Obligation: 0.00/0.72 runtime complexity 0.00/0.72 Answer: 0.00/0.72 YES(O(1),O(n^1)) 0.00/0.72 0.00/0.72 We use the processor 'matrix interpretation of dimension 1' to 0.00/0.72 orient following rules strictly. 0.00/0.72 0.00/0.72 DPs: 0.00/0.72 { 3: D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) } 0.00/0.72 0.00/0.72 Sub-proof: 0.00/0.72 ---------- 0.00/0.72 The following argument positions are usable: 0.00/0.72 Uargs(c_3) = {1, 2}, Uargs(c_4) = {2, 4}, Uargs(c_5) = {1, 2} 0.00/0.72 0.00/0.72 TcT has computed the following constructor-based matrix 0.00/0.72 interpretation satisfying not(EDA). 0.00/0.72 0.00/0.72 [+](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.72 0.00/0.72 [*](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.72 0.00/0.72 [-](x1, x2) = [1] x1 + [1] x2 + [4] 0.00/0.72 0.00/0.72 [D^#](x1) = [2] x1 + [0] 0.00/0.72 0.00/0.72 [c_3](x1, x2) = [1] x1 + [1] x2 + [0] 0.00/0.72 0.00/0.72 [c_4](x1, x2, x3, x4) = [1] x2 + [1] x4 + [0] 0.00/0.72 0.00/0.72 [c_5](x1, x2) = [1] x1 + [1] x2 + [5] 0.00/0.72 0.00/0.72 The order satisfies the following ordering constraints: 0.00/0.72 0.00/0.72 [D^#(+(x, y))] = [2] x + [2] y + [0] 0.00/0.72 >= [2] x + [2] y + [0] 0.00/0.72 = [c_3(D^#(x), D^#(y))] 0.00/0.72 0.00/0.72 [D^#(*(x, y))] = [2] x + [2] y + [0] 0.00/0.72 >= [2] x + [2] y + [0] 0.00/0.72 = [c_4(y, D^#(x), x, D^#(y))] 0.00/0.72 0.00/0.72 [D^#(-(x, y))] = [2] x + [2] y + [8] 0.00/0.72 > [2] x + [2] y + [5] 0.00/0.72 = [c_5(D^#(x), D^#(y))] 0.00/0.72 0.00/0.72 0.00/0.72 The strictly oriented rules are moved into the weak component. 0.00/0.72 0.00/0.72 We are left with following problem, upon which TcT provides the 0.00/0.72 certificate YES(O(1),O(n^1)). 0.00/0.72 0.00/0.72 Strict DPs: 0.00/0.72 { D^#(+(x, y)) -> c_3(D^#(x), D^#(y)) 0.00/0.72 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y)) } 0.00/0.72 Weak DPs: { D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) } 0.00/0.72 Obligation: 0.00/0.72 runtime complexity 0.00/0.72 Answer: 0.00/0.72 YES(O(1),O(n^1)) 0.00/0.72 0.00/0.72 We use the processor 'matrix interpretation of dimension 1' to 0.00/0.72 orient following rules strictly. 0.00/0.72 0.00/0.72 DPs: 0.00/0.72 { 1: D^#(+(x, y)) -> c_3(D^#(x), D^#(y)) 0.00/0.72 , 2: D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y)) 0.00/0.72 , 3: D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) } 0.00/0.72 0.00/0.72 Sub-proof: 0.00/0.72 ---------- 0.00/0.72 The following argument positions are usable: 0.00/0.72 Uargs(c_3) = {1, 2}, Uargs(c_4) = {2, 4}, Uargs(c_5) = {1, 2} 0.00/0.72 0.00/0.72 TcT has computed the following constructor-based matrix 0.00/0.72 interpretation satisfying not(EDA). 0.00/0.72 0.00/0.72 [+](x1, x2) = [1] x1 + [1] x2 + [4] 0.00/0.72 0.00/0.72 [*](x1, x2) = [1] x1 + [1] x2 + [4] 0.00/0.72 0.00/0.72 [-](x1, x2) = [1] x1 + [1] x2 + [4] 0.00/0.72 0.00/0.72 [D^#](x1) = [2] x1 + [0] 0.00/0.72 0.00/0.72 [c_3](x1, x2) = [1] x1 + [1] x2 + [1] 0.00/0.72 0.00/0.72 [c_4](x1, x2, x3, x4) = [1] x2 + [1] x4 + [3] 0.00/0.72 0.00/0.72 [c_5](x1, x2) = [1] x1 + [1] x2 + [7] 0.00/0.72 0.00/0.72 The order satisfies the following ordering constraints: 0.00/0.72 0.00/0.72 [D^#(+(x, y))] = [2] x + [2] y + [8] 0.00/0.72 > [2] x + [2] y + [1] 0.00/0.72 = [c_3(D^#(x), D^#(y))] 0.00/0.72 0.00/0.72 [D^#(*(x, y))] = [2] x + [2] y + [8] 0.00/0.72 > [2] x + [2] y + [3] 0.00/0.72 = [c_4(y, D^#(x), x, D^#(y))] 0.00/0.72 0.00/0.72 [D^#(-(x, y))] = [2] x + [2] y + [8] 0.00/0.72 > [2] x + [2] y + [7] 0.00/0.72 = [c_5(D^#(x), D^#(y))] 0.00/0.72 0.00/0.72 0.00/0.72 The strictly oriented rules are moved into the weak component. 0.00/0.72 0.00/0.72 We are left with following problem, upon which TcT provides the 0.00/0.72 certificate YES(O(1),O(1)). 0.00/0.72 0.00/0.72 Weak DPs: 0.00/0.72 { D^#(+(x, y)) -> c_3(D^#(x), D^#(y)) 0.00/0.72 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y)) 0.00/0.72 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) } 0.00/0.72 Obligation: 0.00/0.72 runtime complexity 0.00/0.72 Answer: 0.00/0.72 YES(O(1),O(1)) 0.00/0.72 0.00/0.72 The following weak DPs constitute a sub-graph of the DG that is 0.00/0.72 closed under successors. The DPs are removed. 0.00/0.72 0.00/0.72 { D^#(+(x, y)) -> c_3(D^#(x), D^#(y)) 0.00/0.72 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y)) 0.00/0.72 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) } 0.00/0.72 0.00/0.72 We are left with following problem, upon which TcT provides the 0.00/0.72 certificate YES(O(1),O(1)). 0.00/0.72 0.00/0.72 Rules: Empty 0.00/0.72 Obligation: 0.00/0.72 runtime complexity 0.00/0.72 Answer: 0.00/0.72 YES(O(1),O(1)) 0.00/0.72 0.00/0.72 Empty rules are trivially bounded 0.00/0.72 0.00/0.72 Hurray, we answered YES(O(1),O(n^1)) 0.00/0.73 EOF