YES(O(1),O(n^2))
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict Trs:
{ *(x, +(y, z)) -> +(*(x, y), *(x, z))
, *(x, 1()) -> x
, *(+(x, y), z) -> +(*(x, z), *(y, z))
, *(1(), y) -> y }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We add the following innermost weak dependency pairs:
Strict DPs:
{ *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
, *^#(x, 1()) -> c_2()
, *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z))
, *^#(1(), y) -> c_4() }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
, *^#(x, 1()) -> c_2()
, *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z))
, *^#(1(), y) -> c_4() }
Strict Trs:
{ *(x, +(y, z)) -> +(*(x, y), *(x, z))
, *(x, 1()) -> x
, *(+(x, y), z) -> +(*(x, z), *(y, z))
, *(1(), y) -> y }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
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^2)).
Strict DPs:
{ *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
, *^#(x, 1()) -> c_2()
, *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z))
, *^#(1(), y) -> c_4() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
The weightgap principle applies (using the following constant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(c_1) = {1, 2}, Uargs(c_3) = {1, 2}
TcT has computed the following constructor-restricted matrix
interpretation.
[+](x1, x2) = [1 2] x1 + [1 2] x2 + [1]
[0 1] [0 1] [1]
[1] = [2]
[1]
[*^#](x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[1 2] [2 2] [1]
[c_1](x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 1] [1]
[c_2] = [0]
[1]
[c_3](x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 1] [1]
[c_4] = [0]
[1]
The following symbols are considered usable
{*^#}
The order satisfies the following ordering constraints:
[*^#(x, +(y, z))] = [0 0] x + [0 0] y + [0 0] z + [1]
[1 2] [2 6] [2 6] [5]
? [0 0] x + [0 0] y + [0 0] z + [3]
[2 4] [2 2] [2 2] [3]
= [c_1(*^#(x, y), *^#(x, z))]
[*^#(x, 1())] = [0 0] x + [1]
[1 2] [7]
> [0]
[1]
= [c_2()]
[*^#(+(x, y), z)] = [0 0] x + [0 0] y + [0 0] z + [1]
[1 4] [1 4] [2 2] [4]
? [0 0] x + [0 0] y + [0 0] z + [3]
[1 2] [1 2] [4 4] [3]
= [c_3(*^#(x, z), *^#(y, z))]
[*^#(1(), y)] = [0 0] y + [1]
[2 2] [5]
> [0]
[1]
= [c_4()]
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^2)).
Strict DPs:
{ *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
, *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) }
Weak DPs:
{ *^#(x, 1()) -> c_2()
, *^#(1(), y) -> c_4() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ *^#(x, 1()) -> c_2()
, *^#(1(), y) -> c_4() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
, *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We use the processor 'polynomial interpretation' to orient
following rules strictly.
DPs:
{ 2: *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) }
Sub-proof:
----------
The following argument positions are considered usable:
Uargs(c_1) = {1, 2}, Uargs(c_3) = {1, 2}
TcT has computed the following constructor-restricted polynomial
interpretation.
[*](x1, x2) = 3*x1 + 3*x1*x2 + 3*x1^2 + 3*x2 + 3*x2^2
[+](x1, x2) = 1 + x1 + x2
[1]() = 0
[*^#](x1, x2) = x1 + x1*x2
[c_1](x1, x2) = x1 + x2
[c_2]() = 0
[c_3](x1, x2) = x1 + x2
[c_4]() = 0
The following symbols are considered usable
{*^#}
This order satisfies the following ordering constraints.
[*^#(x, +(y, z))] = 2*x + x*y + x*z
>= 2*x + x*y + x*z
= [c_1(*^#(x, y), *^#(x, z))]
[*^#(+(x, y), z)] = 1 + x + y + z + x*z + y*z
> x + x*z + y + y*z
= [c_3(*^#(x, z), *^#(y, z))]
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^2)).
Strict DPs: { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z)) }
Weak DPs: { *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We use the processor 'polynomial interpretation' to orient
following rules strictly.
DPs:
{ 1: *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z)) }
Sub-proof:
----------
The following argument positions are considered usable:
Uargs(c_1) = {1, 2}, Uargs(c_3) = {1, 2}
TcT has computed the following constructor-restricted polynomial
interpretation.
[*](x1, x2) = 3*x1 + 3*x1*x2 + 3*x1^2 + 3*x2 + 3*x2^2
[+](x1, x2) = 1 + x1 + x2
[1]() = 0
[*^#](x1, x2) = x1*x2 + x2
[c_1](x1, x2) = x1 + x2
[c_2]() = 0
[c_3](x1, x2) = x1 + x2
[c_4]() = 0
The following symbols are considered usable
{*^#}
This order satisfies the following ordering constraints.
[*^#(x, +(y, z))] = x + x*y + x*z + 1 + y + z
> x*y + y + x*z + z
= [c_1(*^#(x, y), *^#(x, z))]
[*^#(+(x, y), z)] = 2*z + x*z + y*z
>= x*z + 2*z + y*z
= [c_3(*^#(x, z), *^#(y, z))]
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:
{ *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
, *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) }
Obligation:
innermost 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.
{ *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
, *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Rules: Empty
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^2))