YES(O(1),O(n^1))
334.50/148.04 YES(O(1),O(n^1))
334.50/148.04
334.50/148.04 We are left with following problem, upon which TcT provides the
334.50/148.04 certificate YES(O(1),O(n^1)).
334.50/148.04
334.50/148.04 Strict Trs:
334.50/148.04 { D(t()) -> 1()
334.50/148.04 , D(constant()) -> 0()
334.50/148.04 , D(+(x, y)) -> +(D(x), D(y))
334.50/148.04 , D(*(x, y)) -> +(*(y, D(x)), *(x, D(y)))
334.50/148.04 , D(-(x, y)) -> -(D(x), D(y))
334.50/148.04 , D(minus(x)) -> minus(D(x))
334.50/148.04 , D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2())))
334.50/148.04 , D(pow(x, y)) ->
334.50/148.04 +(*(*(y, pow(x, -(y, 1()))), D(x)), *(*(pow(x, y), ln(x)), D(y)))
334.50/148.04 , D(ln(x)) -> div(D(x), x) }
334.50/148.04 Obligation:
334.50/148.04 runtime complexity
334.50/148.04 Answer:
334.50/148.04 YES(O(1),O(n^1))
334.50/148.04
334.50/148.04 We add the following weak dependency pairs:
334.50/148.04
334.50/148.04 Strict DPs:
334.50/148.04 { D^#(t()) -> c_1()
334.50/148.04 , D^#(constant()) -> c_2()
334.50/148.04 , D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
334.50/148.04 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y))
334.50/148.04 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y))
334.50/148.04 , D^#(minus(x)) -> c_6(D^#(x))
334.50/148.04 , D^#(div(x, y)) -> c_7(D^#(x), y, x, D^#(y), y)
334.50/148.04 , D^#(pow(x, y)) -> c_8(y, x, y, D^#(x), x, y, x, D^#(y))
334.50/148.04 , D^#(ln(x)) -> c_9(D^#(x), x) }
334.50/148.04
334.50/148.04 and mark the set of starting terms.
334.50/148.04
334.50/148.04 We are left with following problem, upon which TcT provides the
334.50/148.04 certificate YES(O(1),O(n^1)).
334.50/148.04
334.50/148.04 Strict DPs:
334.50/148.04 { D^#(t()) -> c_1()
334.50/148.04 , D^#(constant()) -> c_2()
334.50/148.04 , D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
334.50/148.04 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y))
334.50/148.04 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y))
334.50/148.04 , D^#(minus(x)) -> c_6(D^#(x))
334.50/148.04 , D^#(div(x, y)) -> c_7(D^#(x), y, x, D^#(y), y)
334.50/148.04 , D^#(pow(x, y)) -> c_8(y, x, y, D^#(x), x, y, x, D^#(y))
334.50/148.04 , D^#(ln(x)) -> c_9(D^#(x), x) }
334.50/148.04 Strict Trs:
334.50/148.04 { D(t()) -> 1()
334.50/148.04 , D(constant()) -> 0()
334.50/148.04 , D(+(x, y)) -> +(D(x), D(y))
334.50/148.04 , D(*(x, y)) -> +(*(y, D(x)), *(x, D(y)))
334.50/148.04 , D(-(x, y)) -> -(D(x), D(y))
334.50/148.04 , D(minus(x)) -> minus(D(x))
334.50/148.04 , D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2())))
334.50/148.04 , D(pow(x, y)) ->
334.50/148.04 +(*(*(y, pow(x, -(y, 1()))), D(x)), *(*(pow(x, y), ln(x)), D(y)))
334.50/148.04 , D(ln(x)) -> div(D(x), x) }
334.50/148.04 Obligation:
334.50/148.04 runtime complexity
334.50/148.04 Answer:
334.50/148.04 YES(O(1),O(n^1))
334.50/148.04
334.50/148.04 No rule is usable, rules are removed from the input problem.
334.50/148.04
334.50/148.04 We are left with following problem, upon which TcT provides the
334.50/148.04 certificate YES(O(1),O(n^1)).
334.50/148.04
334.50/148.04 Strict DPs:
334.50/148.04 { D^#(t()) -> c_1()
334.50/148.04 , D^#(constant()) -> c_2()
334.50/148.04 , D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
334.50/148.04 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y))
334.50/148.04 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y))
334.50/148.04 , D^#(minus(x)) -> c_6(D^#(x))
334.50/148.04 , D^#(div(x, y)) -> c_7(D^#(x), y, x, D^#(y), y)
334.50/148.04 , D^#(pow(x, y)) -> c_8(y, x, y, D^#(x), x, y, x, D^#(y))
334.50/148.04 , D^#(ln(x)) -> c_9(D^#(x), x) }
334.50/148.04 Obligation:
334.50/148.04 runtime complexity
334.50/148.04 Answer:
334.50/148.04 YES(O(1),O(n^1))
334.50/148.04
334.50/148.04 The weightgap principle applies (using the following constant
334.50/148.04 growth matrix-interpretation)
334.50/148.04
334.50/148.04 The following argument positions are usable:
334.50/148.04 Uargs(c_3) = {1, 2}, Uargs(c_4) = {2, 4}, Uargs(c_5) = {1, 2},
334.50/148.04 Uargs(c_6) = {1}, Uargs(c_7) = {1, 4}, Uargs(c_8) = {4, 8},
334.50/148.04 Uargs(c_9) = {1}
334.50/148.04
334.50/148.04 TcT has computed the following constructor-restricted matrix
334.50/148.04 interpretation.
334.50/148.04
334.50/148.04 [t] = [0]
334.50/148.04 [0]
334.50/148.04
334.50/148.04 [constant] = [0]
334.50/148.04 [0]
334.50/148.04
334.50/148.04 [+](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
334.50/148.04 [0 0] [0 0] [0]
334.50/148.04
334.50/148.04 [*](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
334.50/148.04 [0 0] [0 0] [0]
334.50/148.04
334.50/148.04 [-](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
334.50/148.04 [0 0] [0 0] [0]
334.50/148.04
334.50/148.04 [minus](x1) = [1 0] x1 + [0]
334.50/148.04 [0 0] [0]
334.50/148.04
334.50/148.04 [div](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
334.50/148.04 [0 0] [0 0] [0]
334.50/148.04
334.50/148.04 [pow](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
334.50/148.04 [0 0] [0 0] [0]
334.50/148.04
334.50/148.04 [ln](x1) = [1 0] x1 + [0]
334.50/148.04 [0 0] [0]
334.50/148.04
334.50/148.04 [D^#](x1) = [1]
334.50/148.04 [0]
334.50/148.04
334.50/148.04 [c_1] = [0]
334.50/148.04 [0]
334.50/148.04
334.50/148.04 [c_2] = [0]
334.50/148.04 [0]
334.50/148.04
334.50/148.04 [c_3](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
334.50/148.04 [0 1] [0 1] [0]
334.50/148.04
334.50/148.04 [c_4](x1, x2, x3, x4) = [0 0] x1 + [1 0] x2 + [0
334.50/148.04 0] x3 + [1 0] x4 + [2]
334.50/148.04 [2 0] [0 1] [2
334.50/148.04 0] [0 1] [2]
334.50/148.04
334.50/148.04 [c_5](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
334.50/148.04 [0 1] [0 1] [0]
334.50/148.04
334.50/148.04 [c_6](x1) = [1 0] x1 + [1]
334.50/148.04 [0 1] [0]
334.50/148.04
334.50/148.04 [c_7](x1, x2, x3, x4, x5) = [1 0] x1 + [0 0] x2 + [0
334.50/148.04 0] x3 + [1 0] x4 + [0 0] x5 + [2]
334.50/148.04 [0 1] [1 0] [2
334.50/148.04 0] [0 1] [1 0] [0]
334.50/148.04
334.50/148.04 [c_8](x1, x2, x3, x4, x5, x6, x7, x8) = [0 0] x1 + [0 0] x2 + [0
334.50/148.04 0] x3 + [1 0] x4 + [0 0] x5 + [0
334.50/148.04 0] x6 + [0
334.50/148.04 0] x7 + [1
334.50/148.04 0] x8 + [2]
334.50/148.04 [0 2] [1 0] [2
334.50/148.04 0] [0 1] [2 2] [1
334.50/148.04 0] [1
334.50/148.04 2] [0
334.50/148.04 1] [0]
334.50/148.04
334.50/148.04 [c_9](x1, x2) = [1 0] x1 + [0 0] x2 + [1]
334.50/148.04 [0 1] [2 0] [0]
334.50/148.04
334.50/148.04 The order satisfies the following ordering constraints:
334.50/148.04
334.50/148.04 [D^#(t())] = [1]
334.50/148.04 [0]
334.50/148.04 > [0]
334.50/148.04 [0]
334.50/148.04 = [c_1()]
334.50/148.04
334.50/148.04 [D^#(constant())] = [1]
334.50/148.04 [0]
334.50/148.04 > [0]
334.50/148.04 [0]
334.50/148.04 = [c_2()]
334.50/148.04
334.50/148.04 [D^#(+(x, y))] = [1]
334.50/148.04 [0]
334.50/148.04 ? [4]
334.50/148.04 [0]
334.50/148.04 = [c_3(D^#(x), D^#(y))]
334.50/148.04
334.50/148.04 [D^#(*(x, y))] = [1]
334.50/148.04 [0]
334.50/148.04 ? [0 0] x + [0 0] y + [4]
334.50/148.04 [2 0] [2 0] [2]
334.50/148.04 = [c_4(y, D^#(x), x, D^#(y))]
334.50/148.04
334.50/148.04 [D^#(-(x, y))] = [1]
334.50/148.04 [0]
334.50/148.04 ? [4]
334.50/148.04 [0]
334.50/148.04 = [c_5(D^#(x), D^#(y))]
334.50/148.04
334.50/148.04 [D^#(minus(x))] = [1]
334.50/148.04 [0]
334.50/148.04 ? [2]
334.50/148.04 [0]
334.50/148.04 = [c_6(D^#(x))]
334.50/148.04
334.50/148.04 [D^#(div(x, y))] = [1]
334.50/148.04 [0]
334.50/148.04 ? [0 0] x + [0 0] y + [4]
334.50/148.04 [2 0] [2 0] [0]
334.50/148.04 = [c_7(D^#(x), y, x, D^#(y), y)]
334.50/148.04
334.50/148.04 [D^#(pow(x, y))] = [1]
334.50/148.04 [0]
334.50/148.04 ? [0 0] x + [0 0] y + [4]
334.50/148.04 [4 4] [3 2] [0]
334.50/148.04 = [c_8(y, x, y, D^#(x), x, y, x, D^#(y))]
334.50/148.04
334.50/148.04 [D^#(ln(x))] = [1]
334.50/148.04 [0]
334.50/148.04 ? [0 0] x + [2]
334.50/148.04 [2 0] [0]
334.50/148.04 = [c_9(D^#(x), x)]
334.50/148.04
334.50/148.04
334.50/148.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition.
334.50/148.04
334.50/148.04 We are left with following problem, upon which TcT provides the
334.50/148.04 certificate YES(O(1),O(n^1)).
334.50/148.04
334.50/148.04 Strict DPs:
334.50/148.04 { D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
334.50/148.04 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y))
334.50/148.04 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y))
334.50/148.04 , D^#(minus(x)) -> c_6(D^#(x))
334.50/148.04 , D^#(div(x, y)) -> c_7(D^#(x), y, x, D^#(y), y)
334.50/148.04 , D^#(pow(x, y)) -> c_8(y, x, y, D^#(x), x, y, x, D^#(y))
334.50/148.04 , D^#(ln(x)) -> c_9(D^#(x), x) }
334.50/148.04 Weak DPs:
334.50/148.04 { D^#(t()) -> c_1()
334.50/148.04 , D^#(constant()) -> c_2() }
334.50/148.04 Obligation:
334.50/148.04 runtime complexity
334.50/148.04 Answer:
334.50/148.04 YES(O(1),O(n^1))
334.50/148.04
334.50/148.04 The following weak DPs constitute a sub-graph of the DG that is
334.50/148.04 closed under successors. The DPs are removed.
334.50/148.04
334.50/148.04 { D^#(t()) -> c_1()
334.50/148.04 , D^#(constant()) -> c_2() }
334.50/148.04
334.50/148.04 We are left with following problem, upon which TcT provides the
334.50/148.04 certificate YES(O(1),O(n^1)).
334.50/148.04
334.50/148.04 Strict DPs:
334.50/148.04 { D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
334.50/148.04 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y))
334.50/148.04 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y))
334.50/148.04 , D^#(minus(x)) -> c_6(D^#(x))
334.50/148.04 , D^#(div(x, y)) -> c_7(D^#(x), y, x, D^#(y), y)
334.50/148.04 , D^#(pow(x, y)) -> c_8(y, x, y, D^#(x), x, y, x, D^#(y))
334.50/148.04 , D^#(ln(x)) -> c_9(D^#(x), x) }
334.50/148.04 Obligation:
334.50/148.04 runtime complexity
334.50/148.04 Answer:
334.50/148.04 YES(O(1),O(n^1))
334.50/148.04
334.50/148.04 We use the processor 'matrix interpretation of dimension 1' to
334.50/148.04 orient following rules strictly.
334.50/148.05
334.50/148.05 DPs:
334.50/148.05 { 3: D^#(-(x, y)) -> c_5(D^#(x), D^#(y))
334.50/148.05 , 4: D^#(minus(x)) -> c_6(D^#(x))
334.50/148.05 , 5: D^#(div(x, y)) -> c_7(D^#(x), y, x, D^#(y), y)
334.50/148.05 , 6: D^#(pow(x, y)) -> c_8(y, x, y, D^#(x), x, y, x, D^#(y))
334.50/148.05 , 7: D^#(ln(x)) -> c_9(D^#(x), x) }
334.50/148.05
334.50/148.05 Sub-proof:
334.50/148.05 ----------
334.50/148.05 The following argument positions are usable:
334.50/148.05 Uargs(c_3) = {1, 2}, Uargs(c_4) = {2, 4}, Uargs(c_5) = {1, 2},
334.50/148.05 Uargs(c_6) = {1}, Uargs(c_7) = {1, 4}, Uargs(c_8) = {4, 8},
334.50/148.05 Uargs(c_9) = {1}
334.50/148.05
334.50/148.05 TcT has computed the following constructor-based matrix
334.50/148.05 interpretation satisfying not(EDA).
334.50/148.05
334.50/148.05 [+](x1, x2) = [1] x1 + [1] x2 + [0]
334.50/148.05
334.50/148.05 [*](x1, x2) = [1] x1 + [1] x2 + [0]
334.50/148.05
334.50/148.05 [-](x1, x2) = [1] x1 + [1] x2 + [4]
334.50/148.05
334.50/148.05 [minus](x1) = [1] x1 + [4]
334.50/148.05
334.50/148.05 [div](x1, x2) = [1] x1 + [1] x2 + [4]
334.50/148.05
334.50/148.05 [pow](x1, x2) = [1] x1 + [1] x2 + [4]
334.50/148.05
334.50/148.05 [ln](x1) = [1] x1 + [4]
334.50/148.05
334.50/148.05 [D^#](x1) = [2] x1 + [0]
334.50/148.05
334.50/148.05 [c_3](x1, x2) = [1] x1 + [1] x2 + [0]
334.50/148.05
334.50/148.05 [c_4](x1, x2, x3, x4) = [1] x2 + [1] x4 + [0]
334.50/148.05
334.50/148.05 [c_5](x1, x2) = [1] x1 + [1] x2 + [7]
334.50/148.05
334.50/148.05 [c_6](x1) = [1] x1 + [3]
334.50/148.05
334.50/148.05 [c_7](x1, x2, x3, x4, x5) = [1] x1 + [1] x4 + [7]
334.50/148.05
334.50/148.05 [c_8](x1, x2, x3, x4, x5, x6, x7, x8) = [1] x4 + [1] x8 + [1]
334.50/148.05
334.50/148.05 [c_9](x1, x2) = [1] x1 + [3]
334.50/148.05
334.50/148.05 The order satisfies the following ordering constraints:
334.50/148.05
334.50/148.05 [D^#(+(x, y))] = [2] x + [2] y + [0]
334.50/148.05 >= [2] x + [2] y + [0]
334.50/148.05 = [c_3(D^#(x), D^#(y))]
334.50/148.05
334.50/148.05 [D^#(*(x, y))] = [2] x + [2] y + [0]
334.50/148.05 >= [2] x + [2] y + [0]
334.50/148.05 = [c_4(y, D^#(x), x, D^#(y))]
334.50/148.05
334.50/148.05 [D^#(-(x, y))] = [2] x + [2] y + [8]
334.50/148.05 > [2] x + [2] y + [7]
334.50/148.05 = [c_5(D^#(x), D^#(y))]
334.50/148.05
334.50/148.05 [D^#(minus(x))] = [2] x + [8]
334.50/148.05 > [2] x + [3]
334.50/148.05 = [c_6(D^#(x))]
334.50/148.05
334.50/148.05 [D^#(div(x, y))] = [2] x + [2] y + [8]
334.50/148.05 > [2] x + [2] y + [7]
334.50/148.05 = [c_7(D^#(x), y, x, D^#(y), y)]
334.50/148.05
334.50/148.05 [D^#(pow(x, y))] = [2] x + [2] y + [8]
334.50/148.05 > [2] x + [2] y + [1]
334.50/148.05 = [c_8(y, x, y, D^#(x), x, y, x, D^#(y))]
334.50/148.05
334.50/148.05 [D^#(ln(x))] = [2] x + [8]
334.50/148.05 > [2] x + [3]
334.50/148.05 = [c_9(D^#(x), x)]
334.50/148.05
334.50/148.05
334.50/148.05 The strictly oriented rules are moved into the weak component.
334.50/148.05
334.50/148.05 We are left with following problem, upon which TcT provides the
334.50/148.05 certificate YES(O(1),O(n^1)).
334.50/148.05
334.50/148.05 Strict DPs:
334.50/148.05 { D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
334.50/148.05 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y)) }
334.50/148.05 Weak DPs:
334.50/148.05 { D^#(-(x, y)) -> c_5(D^#(x), D^#(y))
334.50/148.05 , D^#(minus(x)) -> c_6(D^#(x))
334.50/148.05 , D^#(div(x, y)) -> c_7(D^#(x), y, x, D^#(y), y)
334.50/148.05 , D^#(pow(x, y)) -> c_8(y, x, y, D^#(x), x, y, x, D^#(y))
334.50/148.05 , D^#(ln(x)) -> c_9(D^#(x), x) }
334.50/148.05 Obligation:
334.50/148.05 runtime complexity
334.50/148.05 Answer:
334.50/148.05 YES(O(1),O(n^1))
334.50/148.05
334.50/148.05 We use the processor 'matrix interpretation of dimension 1' to
334.50/148.05 orient following rules strictly.
334.50/148.05
334.50/148.05 DPs:
334.50/148.05 { 1: D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
334.50/148.05 , 2: D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y))
334.50/148.05 , 3: D^#(-(x, y)) -> c_5(D^#(x), D^#(y))
334.50/148.05 , 4: D^#(minus(x)) -> c_6(D^#(x))
334.50/148.05 , 5: D^#(div(x, y)) -> c_7(D^#(x), y, x, D^#(y), y)
334.50/148.05 , 6: D^#(pow(x, y)) -> c_8(y, x, y, D^#(x), x, y, x, D^#(y))
334.50/148.05 , 7: D^#(ln(x)) -> c_9(D^#(x), x) }
334.50/148.05
334.50/148.05 Sub-proof:
334.50/148.05 ----------
334.50/148.05 The following argument positions are usable:
334.50/148.05 Uargs(c_3) = {1, 2}, Uargs(c_4) = {2, 4}, Uargs(c_5) = {1, 2},
334.50/148.05 Uargs(c_6) = {1}, Uargs(c_7) = {1, 4}, Uargs(c_8) = {4, 8},
334.50/148.05 Uargs(c_9) = {1}
334.50/148.05
334.50/148.05 TcT has computed the following constructor-based matrix
334.50/148.05 interpretation satisfying not(EDA).
334.50/148.05
334.50/148.05 [+](x1, x2) = [1] x1 + [1] x2 + [4]
334.50/148.05
334.50/148.05 [*](x1, x2) = [1] x1 + [1] x2 + [4]
334.50/148.05
334.50/148.05 [-](x1, x2) = [1] x1 + [1] x2 + [4]
334.50/148.05
334.50/148.05 [minus](x1) = [1] x1 + [4]
334.50/148.05
334.50/148.05 [div](x1, x2) = [1] x1 + [1] x2 + [4]
334.50/148.05
334.50/148.05 [pow](x1, x2) = [1] x1 + [1] x2 + [4]
334.50/148.05
334.50/148.05 [ln](x1) = [1] x1 + [4]
334.50/148.05
334.50/148.05 [D^#](x1) = [2] x1 + [0]
334.50/148.05
334.50/148.05 [c_3](x1, x2) = [1] x1 + [1] x2 + [7]
334.50/148.05
334.50/148.05 [c_4](x1, x2, x3, x4) = [1] x2 + [1] x4 + [1]
334.50/148.05
334.50/148.05 [c_5](x1, x2) = [1] x1 + [1] x2 + [3]
334.50/148.05
334.50/148.05 [c_6](x1) = [1] x1 + [5]
334.50/148.05
334.50/148.05 [c_7](x1, x2, x3, x4, x5) = [1] x1 + [1] x4 + [7]
334.50/148.05
334.50/148.05 [c_8](x1, x2, x3, x4, x5, x6, x7, x8) = [1] x4 + [1] x8 + [7]
334.50/148.05
334.50/148.05 [c_9](x1, x2) = [1] x1 + [5]
334.50/148.05
334.50/148.05 The order satisfies the following ordering constraints:
334.50/148.05
334.50/148.05 [D^#(+(x, y))] = [2] x + [2] y + [8]
334.50/148.05 > [2] x + [2] y + [7]
334.50/148.05 = [c_3(D^#(x), D^#(y))]
334.50/148.05
334.50/148.05 [D^#(*(x, y))] = [2] x + [2] y + [8]
334.50/148.05 > [2] x + [2] y + [1]
334.50/148.05 = [c_4(y, D^#(x), x, D^#(y))]
334.50/148.05
334.50/148.05 [D^#(-(x, y))] = [2] x + [2] y + [8]
334.50/148.05 > [2] x + [2] y + [3]
334.50/148.05 = [c_5(D^#(x), D^#(y))]
334.50/148.05
334.50/148.05 [D^#(minus(x))] = [2] x + [8]
334.50/148.05 > [2] x + [5]
334.50/148.05 = [c_6(D^#(x))]
334.50/148.05
334.50/148.05 [D^#(div(x, y))] = [2] x + [2] y + [8]
334.50/148.05 > [2] x + [2] y + [7]
334.50/148.05 = [c_7(D^#(x), y, x, D^#(y), y)]
334.50/148.05
334.50/148.05 [D^#(pow(x, y))] = [2] x + [2] y + [8]
334.50/148.05 > [2] x + [2] y + [7]
334.50/148.05 = [c_8(y, x, y, D^#(x), x, y, x, D^#(y))]
334.50/148.05
334.50/148.05 [D^#(ln(x))] = [2] x + [8]
334.50/148.05 > [2] x + [5]
334.50/148.05 = [c_9(D^#(x), x)]
334.50/148.05
334.50/148.05
334.50/148.05 The strictly oriented rules are moved into the weak component.
334.50/148.05
334.50/148.05 We are left with following problem, upon which TcT provides the
334.50/148.05 certificate YES(O(1),O(1)).
334.50/148.05
334.50/148.05 Weak DPs:
334.50/148.05 { D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
334.50/148.05 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y))
334.50/148.05 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y))
334.50/148.05 , D^#(minus(x)) -> c_6(D^#(x))
334.50/148.05 , D^#(div(x, y)) -> c_7(D^#(x), y, x, D^#(y), y)
334.50/148.05 , D^#(pow(x, y)) -> c_8(y, x, y, D^#(x), x, y, x, D^#(y))
334.50/148.05 , D^#(ln(x)) -> c_9(D^#(x), x) }
334.50/148.05 Obligation:
334.50/148.05 runtime complexity
334.50/148.05 Answer:
334.50/148.05 YES(O(1),O(1))
334.50/148.05
334.50/148.05 The following weak DPs constitute a sub-graph of the DG that is
334.50/148.05 closed under successors. The DPs are removed.
334.50/148.05
334.50/148.05 { D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
334.50/148.05 , D^#(*(x, y)) -> c_4(y, D^#(x), x, D^#(y))
334.50/148.05 , D^#(-(x, y)) -> c_5(D^#(x), D^#(y))
334.50/148.05 , D^#(minus(x)) -> c_6(D^#(x))
334.50/148.05 , D^#(div(x, y)) -> c_7(D^#(x), y, x, D^#(y), y)
334.50/148.05 , D^#(pow(x, y)) -> c_8(y, x, y, D^#(x), x, y, x, D^#(y))
334.50/148.05 , D^#(ln(x)) -> c_9(D^#(x), x) }
334.50/148.05
334.50/148.05 We are left with following problem, upon which TcT provides the
334.50/148.05 certificate YES(O(1),O(1)).
334.50/148.05
334.50/148.05 Rules: Empty
334.50/148.05 Obligation:
334.50/148.05 runtime complexity
334.50/148.05 Answer:
334.50/148.05 YES(O(1),O(1))
334.50/148.05
334.50/148.05 Empty rules are trivially bounded
334.50/148.05
334.50/148.05 Hurray, we answered YES(O(1),O(n^1))
334.82/148.20 EOF