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