YES(O(1),O(n^1)) 42.40/12.57 YES(O(1),O(n^1)) 42.40/12.57 42.40/12.57 We are left with following problem, upon which TcT provides the 42.40/12.57 certificate YES(O(1),O(n^1)). 42.40/12.57 42.40/12.57 Strict Trs: 42.40/12.57 { f(c(a(), z, x)) -> b(a(), z) 42.40/12.57 , b(y, z) -> z 42.40/12.57 , b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y))) } 42.40/12.57 Obligation: 42.40/12.57 runtime complexity 42.40/12.57 Answer: 42.40/12.57 YES(O(1),O(n^1)) 42.40/12.57 42.40/12.57 We add the following weak dependency pairs: 42.40/12.57 42.40/12.57 Strict DPs: 42.40/12.57 { f^#(c(a(), z, x)) -> c_1(b^#(a(), z)) 42.40/12.57 , b^#(y, z) -> c_2(z) 42.40/12.57 , b^#(x, b(z, y)) -> c_3(f^#(b(f(f(z)), c(x, z, y)))) } 42.40/12.57 42.40/12.57 and mark the set of starting terms. 42.40/12.57 42.40/12.57 We are left with following problem, upon which TcT provides the 42.40/12.57 certificate YES(O(1),O(n^1)). 42.40/12.57 42.40/12.57 Strict DPs: 42.40/12.57 { f^#(c(a(), z, x)) -> c_1(b^#(a(), z)) 42.40/12.57 , b^#(y, z) -> c_2(z) 42.40/12.57 , b^#(x, b(z, y)) -> c_3(f^#(b(f(f(z)), c(x, z, y)))) } 42.40/12.57 Strict Trs: 42.40/12.57 { f(c(a(), z, x)) -> b(a(), z) 42.40/12.57 , b(y, z) -> z 42.40/12.57 , b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y))) } 42.40/12.57 Obligation: 42.40/12.57 runtime complexity 42.40/12.57 Answer: 42.40/12.57 YES(O(1),O(n^1)) 42.40/12.57 42.40/12.57 The weightgap principle applies (using the following constant 42.40/12.57 growth matrix-interpretation) 42.40/12.57 42.40/12.57 The following argument positions are usable: 42.40/12.57 Uargs(f) = {1}, Uargs(c) = {1, 2, 3}, Uargs(b) = {1, 2}, 42.40/12.57 Uargs(f^#) = {1}, Uargs(c_1) = {1}, Uargs(b^#) = {2}, 42.40/12.57 Uargs(c_2) = {1}, Uargs(c_3) = {1} 42.40/12.57 42.40/12.57 TcT has computed the following constructor-restricted matrix 42.40/12.57 interpretation. 42.40/12.57 42.40/12.57 [f](x1) = [1 0] x1 + [0] 42.40/12.57 [2 0] [0] 42.40/12.57 42.40/12.57 [c](x1, x2, x3) = [1 0] x1 + [1 2] x2 + [1 0] x3 + [2] 42.40/12.57 [0 0] [0 0] [0 0] [0] 42.40/12.57 42.40/12.57 [a] = [0] 42.40/12.57 [0] 42.40/12.57 42.40/12.57 [b](x1, x2) = [1 0] x1 + [1 1] x2 + [1] 42.40/12.57 [2 2] [1 2] [2] 42.40/12.57 42.40/12.57 [f^#](x1) = [1 0] x1 + [1] 42.40/12.57 [0 0] [2] 42.40/12.57 42.40/12.57 [c_1](x1) = [1 0] x1 + [2] 42.40/12.57 [0 1] [1] 42.40/12.57 42.40/12.57 [b^#](x1, x2) = [2 2] x1 + [1 1] x2 + [2] 42.40/12.57 [1 2] [1 2] [1] 42.40/12.57 42.40/12.57 [c_2](x1) = [1 0] x1 + [1] 42.40/12.57 [0 1] [1] 42.40/12.57 42.40/12.57 [c_3](x1) = [1 0] x1 + [2] 42.40/12.57 [0 1] [2] 42.40/12.57 42.40/12.57 The order satisfies the following ordering constraints: 42.40/12.57 42.40/12.57 [f(c(a(), z, x))] = [1 2] z + [1 0] x + [2] 42.40/12.57 [2 4] [2 0] [4] 42.40/12.57 > [1 1] z + [1] 42.40/12.57 [1 2] [2] 42.40/12.57 = [b(a(), z)] 42.40/12.57 42.40/12.57 [b(y, z)] = [1 1] z + [1 0] y + [1] 42.40/12.57 [1 2] [2 2] [2] 42.40/12.57 > [1 0] z + [0] 42.40/12.57 [0 1] [0] 42.40/12.57 = [z] 42.40/12.57 42.40/12.57 [b(x, b(z, y))] = [3 2] z + [1 0] x + [2 3] y + [4] 42.40/12.57 [5 4] [2 2] [3 5] [7] 42.40/12.57 > [2 2] z + [1 0] x + [1 0] y + [3] 42.40/12.57 [4 4] [2 0] [2 0] [6] 42.40/12.57 = [f(b(f(f(z)), c(x, z, y)))] 42.40/12.57 42.40/12.57 [f^#(c(a(), z, x))] = [1 2] z + [1 0] x + [3] 42.40/12.57 [0 0] [0 0] [2] 42.40/12.57 ? [1 1] z + [4] 42.40/12.57 [1 2] [2] 42.40/12.57 = [c_1(b^#(a(), z))] 42.40/12.57 42.40/12.57 [b^#(y, z)] = [1 1] z + [2 2] y + [2] 42.40/12.57 [1 2] [1 2] [1] 42.40/12.57 > [1 0] z + [1] 42.40/12.57 [0 1] [1] 42.40/12.57 = [c_2(z)] 42.40/12.57 42.40/12.57 [b^#(x, b(z, y))] = [3 2] z + [2 2] x + [2 3] y + [5] 42.40/12.57 [5 4] [1 2] [3 5] [6] 42.40/12.57 ? [2 2] z + [1 0] x + [1 0] y + [6] 42.40/12.57 [0 0] [0 0] [0 0] [4] 42.40/12.57 = [c_3(f^#(b(f(f(z)), c(x, z, y))))] 42.40/12.57 42.40/12.57 42.40/12.57 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 42.40/12.57 42.40/12.57 We are left with following problem, upon which TcT provides the 42.40/12.57 certificate YES(O(1),O(n^1)). 42.40/12.57 42.40/12.57 Strict DPs: 42.40/12.57 { f^#(c(a(), z, x)) -> c_1(b^#(a(), z)) 42.40/12.57 , b^#(x, b(z, y)) -> c_3(f^#(b(f(f(z)), c(x, z, y)))) } 42.40/12.57 Weak DPs: { b^#(y, z) -> c_2(z) } 42.40/12.57 Weak Trs: 42.40/12.57 { f(c(a(), z, x)) -> b(a(), z) 42.40/12.57 , b(y, z) -> z 42.40/12.57 , b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y))) } 42.40/12.57 Obligation: 42.40/12.57 runtime complexity 42.40/12.57 Answer: 42.40/12.57 YES(O(1),O(n^1)) 42.40/12.57 42.40/12.57 We use the processor 'matrix interpretation of dimension 2' to 42.40/12.57 orient following rules strictly. 42.40/12.57 42.40/12.57 DPs: 42.40/12.57 { 2: b^#(x, b(z, y)) -> c_3(f^#(b(f(f(z)), c(x, z, y)))) } 42.40/12.57 Trs: 42.40/12.57 { f(c(a(), z, x)) -> b(a(), z) 42.40/12.57 , b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y))) } 42.40/12.57 42.40/12.57 Sub-proof: 42.40/12.57 ---------- 42.40/12.57 The following argument positions are usable: 42.40/12.57 Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1} 42.40/12.57 42.40/12.57 TcT has computed the following constructor-based matrix 42.40/12.57 interpretation satisfying not(EDA) and not(IDA(1)). 42.40/12.57 42.40/12.57 [f](x1) = [1 0] x1 + [0] 42.40/12.57 [2 0] [0] 42.40/12.57 42.40/12.57 [c](x1, x2, x3) = [1 4] x2 + [1] 42.40/12.57 [0 0] [0] 42.40/12.57 42.40/12.57 [a] = [0] 42.40/12.57 [0] 42.40/12.57 42.40/12.57 [b](x1, x2) = [0 2] x1 + [1 3] x2 + [0] 42.40/12.57 [5 2] [2 2] [1] 42.40/12.57 42.40/12.57 [f^#](x1) = [1 0] x1 + [0] 42.40/12.57 [2 0] [0] 42.40/12.57 42.40/12.57 [c_1](x1) = [1 0] x1 + [1] 42.40/12.57 [0 0] [1] 42.40/12.57 42.40/12.57 [b^#](x1, x2) = [4 4] x1 + [1 2] x2 + [0] 42.40/12.57 [0 4] [0 0] [0] 42.40/12.57 42.40/12.57 [c_2](x1) = [1 1] x1 + [0] 42.40/12.57 [0 0] [0] 42.40/12.57 42.40/12.57 [c_3](x1) = [1 0] x1 + [0] 42.40/12.57 [0 0] [0] 42.40/12.57 42.40/12.57 The order satisfies the following ordering constraints: 42.40/12.57 42.40/12.57 [f(c(a(), z, x))] = [1 4] z + [1] 42.40/12.57 [2 8] [2] 42.40/12.57 > [1 3] z + [0] 42.40/12.57 [2 2] [1] 42.40/12.57 = [b(a(), z)] 42.40/12.57 42.40/12.57 [b(y, z)] = [1 3] z + [0 2] y + [0] 42.40/12.57 [2 2] [5 2] [1] 42.40/12.57 >= [1 0] z + [0] 42.40/12.57 [0 1] [0] 42.40/12.57 = [z] 42.40/12.57 42.40/12.57 [b(x, b(z, y))] = [15 8] z + [0 2] x + [7 9] y + [3] 42.40/12.57 [10 8] [5 2] [6 10] [3] 42.40/12.58 > [ 5 4] z + [1] 42.40/12.58 [10 8] [2] 42.40/12.58 = [f(b(f(f(z)), c(x, z, y)))] 42.40/12.58 42.40/12.58 [f^#(c(a(), z, x))] = [1 4] z + [1] 42.40/12.58 [2 8] [2] 42.40/12.58 >= [1 2] z + [1] 42.40/12.58 [0 0] [1] 42.40/12.58 = [c_1(b^#(a(), z))] 42.40/12.58 42.40/12.58 [b^#(y, z)] = [1 2] z + [4 4] y + [0] 42.40/12.58 [0 0] [0 4] [0] 42.40/12.58 >= [1 1] z + [0] 42.40/12.58 [0 0] [0] 42.40/12.58 = [c_2(z)] 42.40/12.58 42.40/12.58 [b^#(x, b(z, y))] = [10 6] z + [4 4] x + [5 7] y + [2] 42.40/12.58 [ 0 0] [0 4] [0 0] [0] 42.40/12.58 > [5 4] z + [1] 42.40/12.58 [0 0] [0] 42.40/12.58 = [c_3(f^#(b(f(f(z)), c(x, z, y))))] 42.40/12.58 42.40/12.58 42.40/12.58 The strictly oriented rules are moved into the weak component. 42.40/12.58 42.40/12.58 We are left with following problem, upon which TcT provides the 42.40/12.58 certificate YES(O(1),O(n^1)). 42.40/12.58 42.40/12.58 Strict DPs: { f^#(c(a(), z, x)) -> c_1(b^#(a(), z)) } 42.40/12.58 Weak DPs: 42.40/12.58 { b^#(y, z) -> c_2(z) 42.40/12.58 , b^#(x, b(z, y)) -> c_3(f^#(b(f(f(z)), c(x, z, y)))) } 42.40/12.58 Weak Trs: 42.40/12.58 { f(c(a(), z, x)) -> b(a(), z) 42.40/12.58 , b(y, z) -> z 42.40/12.58 , b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y))) } 42.40/12.58 Obligation: 42.40/12.58 runtime complexity 42.40/12.58 Answer: 42.40/12.58 YES(O(1),O(n^1)) 42.40/12.58 42.40/12.58 We use the processor 'matrix interpretation of dimension 2' to 42.40/12.58 orient following rules strictly. 42.40/12.58 42.40/12.58 DPs: 42.40/12.58 { 1: f^#(c(a(), z, x)) -> c_1(b^#(a(), z)) } 42.40/12.58 Trs: 42.40/12.58 { b(y, z) -> z 42.40/12.58 , b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y))) } 42.40/12.58 42.40/12.58 Sub-proof: 42.40/12.58 ---------- 42.40/12.58 The following argument positions are usable: 42.40/12.58 Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1} 42.40/12.58 42.40/12.58 TcT has computed the following constructor-based matrix 42.40/12.58 interpretation satisfying not(EDA) and not(IDA(1)). 42.40/12.58 42.40/12.58 [f](x1) = [0 2] x1 + [2] 42.40/12.58 [0 2] [0] 42.40/12.58 42.40/12.58 [c](x1, x2, x3) = [0 0] x2 + [0] 42.40/12.58 [1 1] [0] 42.40/12.58 42.40/12.58 [a] = [0] 42.40/12.58 [0] 42.40/12.58 42.40/12.58 [b](x1, x2) = [4 4] x1 + [2 0] x2 + [2] 42.40/12.58 [0 0] [1 2] [0] 42.40/12.58 42.40/12.58 [f^#](x1) = [0 1] x1 + [1] 42.40/12.58 [0 4] [4] 42.40/12.58 42.40/12.58 [c_1](x1) = [1 0] x1 + [0] 42.40/12.58 [0 0] [3] 42.40/12.58 42.40/12.58 [b^#](x1, x2) = [4 0] x1 + [1 0] x2 + [0] 42.40/12.58 [0 0] [0 0] [0] 42.40/12.58 42.40/12.58 [c_2](x1) = [1 0] x1 + [0] 42.40/12.58 [0 0] [0] 42.40/12.58 42.40/12.58 [c_3](x1) = [2 0] x1 + [0] 42.40/12.58 [0 0] [0] 42.40/12.58 42.40/12.58 The order satisfies the following ordering constraints: 42.40/12.58 42.40/12.58 [f(c(a(), z, x))] = [2 2] z + [2] 42.40/12.58 [2 2] [0] 42.40/12.58 >= [2 0] z + [2] 42.40/12.58 [1 2] [0] 42.40/12.58 = [b(a(), z)] 42.40/12.58 42.40/12.58 [b(y, z)] = [2 0] z + [4 4] y + [2] 42.40/12.58 [1 2] [0 0] [0] 42.40/12.58 > [1 0] z + [0] 42.40/12.58 [0 1] [0] 42.40/12.58 = [z] 42.40/12.58 42.40/12.58 [b(x, b(z, y))] = [8 8] z + [4 4] x + [4 0] y + [6] 42.40/12.58 [4 4] [0 0] [4 4] [2] 42.40/12.58 > [4 4] z + [2] 42.40/12.58 [4 4] [0] 42.40/12.58 = [f(b(f(f(z)), c(x, z, y)))] 42.40/12.58 42.40/12.58 [f^#(c(a(), z, x))] = [1 1] z + [1] 42.40/12.58 [4 4] [4] 42.40/12.58 > [1 0] z + [0] 42.40/12.58 [0 0] [3] 42.40/12.58 = [c_1(b^#(a(), z))] 42.40/12.58 42.40/12.58 [b^#(y, z)] = [1 0] z + [4 0] y + [0] 42.40/12.58 [0 0] [0 0] [0] 42.40/12.58 >= [1 0] z + [0] 42.40/12.58 [0 0] [0] 42.40/12.58 = [c_2(z)] 42.40/12.58 42.40/12.58 [b^#(x, b(z, y))] = [4 4] z + [4 0] x + [2 0] y + [2] 42.40/12.58 [0 0] [0 0] [0 0] [0] 42.40/12.58 >= [4 4] z + [2] 42.40/12.58 [0 0] [0] 42.40/12.58 = [c_3(f^#(b(f(f(z)), c(x, z, y))))] 42.40/12.58 42.40/12.58 42.40/12.58 The strictly oriented rules are moved into the weak component. 42.40/12.58 42.40/12.58 We are left with following problem, upon which TcT provides the 42.40/12.58 certificate YES(O(1),O(1)). 42.40/12.58 42.40/12.58 Weak DPs: 42.40/12.58 { f^#(c(a(), z, x)) -> c_1(b^#(a(), z)) 42.40/12.58 , b^#(y, z) -> c_2(z) 42.40/12.58 , b^#(x, b(z, y)) -> c_3(f^#(b(f(f(z)), c(x, z, y)))) } 42.40/12.58 Weak Trs: 42.40/12.58 { f(c(a(), z, x)) -> b(a(), z) 42.40/12.58 , b(y, z) -> z 42.40/12.58 , b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y))) } 42.40/12.58 Obligation: 42.40/12.58 runtime complexity 42.40/12.58 Answer: 42.40/12.58 YES(O(1),O(1)) 42.40/12.58 42.40/12.58 The following weak DPs constitute a sub-graph of the DG that is 42.40/12.58 closed under successors. The DPs are removed. 42.40/12.58 42.40/12.58 { f^#(c(a(), z, x)) -> c_1(b^#(a(), z)) 42.40/12.58 , b^#(y, z) -> c_2(z) 42.40/12.58 , b^#(x, b(z, y)) -> c_3(f^#(b(f(f(z)), c(x, z, y)))) } 42.40/12.58 42.40/12.58 We are left with following problem, upon which TcT provides the 42.40/12.58 certificate YES(O(1),O(1)). 42.40/12.58 42.40/12.58 Weak Trs: 42.40/12.58 { f(c(a(), z, x)) -> b(a(), z) 42.40/12.58 , b(y, z) -> z 42.40/12.58 , b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y))) } 42.40/12.58 Obligation: 42.40/12.58 runtime complexity 42.40/12.58 Answer: 42.40/12.58 YES(O(1),O(1)) 42.40/12.58 42.40/12.58 No rule is usable, rules are removed from the input problem. 42.40/12.58 42.40/12.58 We are left with following problem, upon which TcT provides the 42.40/12.58 certificate YES(O(1),O(1)). 42.40/12.58 42.40/12.58 Rules: Empty 42.40/12.58 Obligation: 42.40/12.58 runtime complexity 42.40/12.58 Answer: 42.40/12.58 YES(O(1),O(1)) 42.40/12.58 42.40/12.58 Empty rules are trivially bounded 42.40/12.58 42.40/12.58 Hurray, we answered YES(O(1),O(n^1)) 42.63/12.61 EOF