We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ perfectp(0()) -> false()
, perfectp(s(x)) -> f(x, s(0()), s(x), s(x))
, f(0(), y, 0(), u) -> true()
, f(0(), y, s(z), u) -> false()
, f(s(x), 0(), z, u) -> f(x, u, minus(z, s(x)), u)
, f(s(x), s(y), z, u) ->
if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u)) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ perfectp^#(0()) -> c_1()
, perfectp^#(s(x)) -> c_2(f^#(x, s(0()), s(x), s(x)))
, f^#(0(), y, 0(), u) -> c_3()
, f^#(0(), y, s(z), u) -> c_4()
, f^#(s(x), 0(), z, u) -> c_5(f^#(x, u, minus(z, s(x)), u))
, f^#(s(x), s(y), z, u) ->
c_6(x, y, f^#(s(x), minus(y, x), z, u), f^#(x, u, z, u)) }
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:
{ perfectp^#(0()) -> c_1()
, perfectp^#(s(x)) -> c_2(f^#(x, s(0()), s(x), s(x)))
, f^#(0(), y, 0(), u) -> c_3()
, f^#(0(), y, s(z), u) -> c_4()
, f^#(s(x), 0(), z, u) -> c_5(f^#(x, u, minus(z, s(x)), u))
, f^#(s(x), s(y), z, u) ->
c_6(x, y, f^#(s(x), minus(y, x), z, u), f^#(x, u, z, u)) }
Strict Trs:
{ perfectp(0()) -> false()
, perfectp(s(x)) -> f(x, s(0()), s(x), s(x))
, f(0(), y, 0(), u) -> true()
, f(0(), y, s(z), u) -> false()
, f(s(x), 0(), z, u) -> f(x, u, minus(z, s(x)), u)
, f(s(x), s(y), z, u) ->
if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u)) }
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:
{ perfectp^#(0()) -> c_1()
, perfectp^#(s(x)) -> c_2(f^#(x, s(0()), s(x), s(x)))
, f^#(0(), y, 0(), u) -> c_3()
, f^#(0(), y, s(z), u) -> c_4()
, f^#(s(x), 0(), z, u) -> c_5(f^#(x, u, minus(z, s(x)), u))
, f^#(s(x), s(y), z, u) ->
c_6(x, y, f^#(s(x), minus(y, x), z, u), f^#(x, u, z, u)) }
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_2) = {1}, Uargs(c_5) = {1}, Uargs(c_6) = {4}
TcT has computed the following constructor-restricted matrix
interpretation.
[0] = [0]
[0]
[s](x1) = [0]
[0]
[minus](x1, x2) = [0]
[0]
[perfectp^#](x1) = [0]
[0]
[c_1] = [0]
[0]
[c_2](x1) = [1 0] x1 + [0]
[0 1] [0]
[f^#](x1, x2, x3, x4) = [1]
[0]
[c_3] = [0]
[0]
[c_4] = [0]
[0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6](x1, x2, x3, x4) = [0 0] x2 + [1 0] x4 + [0]
[2 0] [0 1] [0]
The order satisfies the following ordering constraints:
[perfectp^#(0())] = [0]
[0]
>= [0]
[0]
= [c_1()]
[perfectp^#(s(x))] = [0]
[0]
? [1]
[0]
= [c_2(f^#(x, s(0()), s(x), s(x)))]
[f^#(0(), y, 0(), u)] = [1]
[0]
> [0]
[0]
= [c_3()]
[f^#(0(), y, s(z), u)] = [1]
[0]
> [0]
[0]
= [c_4()]
[f^#(s(x), 0(), z, u)] = [1]
[0]
>= [1]
[0]
= [c_5(f^#(x, u, minus(z, s(x)), u))]
[f^#(s(x), s(y), z, u)] = [1]
[0]
? [0 0] y + [1]
[2 0] [0]
= [c_6(x, y, f^#(s(x), minus(y, x), z, u), f^#(x, u, z, u))]
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:
{ perfectp^#(0()) -> c_1()
, perfectp^#(s(x)) -> c_2(f^#(x, s(0()), s(x), s(x)))
, f^#(s(x), 0(), z, u) -> c_5(f^#(x, u, minus(z, s(x)), u))
, f^#(s(x), s(y), z, u) ->
c_6(x, y, f^#(s(x), minus(y, x), z, u), f^#(x, u, z, u)) }
Weak DPs:
{ f^#(0(), y, 0(), u) -> c_3()
, f^#(0(), y, s(z), u) -> c_4() }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We estimate the number of application of {1} by applications of
Pre({1}) = {4}. Here rules are labeled as follows:
DPs:
{ 1: perfectp^#(0()) -> c_1()
, 2: perfectp^#(s(x)) -> c_2(f^#(x, s(0()), s(x), s(x)))
, 3: f^#(s(x), 0(), z, u) -> c_5(f^#(x, u, minus(z, s(x)), u))
, 4: f^#(s(x), s(y), z, u) ->
c_6(x, y, f^#(s(x), minus(y, x), z, u), f^#(x, u, z, u))
, 5: f^#(0(), y, 0(), u) -> c_3()
, 6: f^#(0(), y, s(z), u) -> c_4() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ perfectp^#(s(x)) -> c_2(f^#(x, s(0()), s(x), s(x)))
, f^#(s(x), 0(), z, u) -> c_5(f^#(x, u, minus(z, s(x)), u))
, f^#(s(x), s(y), z, u) ->
c_6(x, y, f^#(s(x), minus(y, x), z, u), f^#(x, u, z, u)) }
Weak DPs:
{ perfectp^#(0()) -> c_1()
, f^#(0(), y, 0(), u) -> c_3()
, f^#(0(), y, s(z), u) -> c_4() }
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.
{ perfectp^#(0()) -> c_1()
, f^#(0(), y, 0(), u) -> c_3()
, f^#(0(), y, s(z), u) -> c_4() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ perfectp^#(s(x)) -> c_2(f^#(x, s(0()), s(x), s(x)))
, f^#(s(x), 0(), z, u) -> c_5(f^#(x, u, minus(z, s(x)), u))
, f^#(s(x), s(y), z, u) ->
c_6(x, y, f^#(s(x), minus(y, x), z, u), f^#(x, u, z, u)) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:
{ f^#(s(x), s(y), z, u) ->
c_6(x, y, f^#(s(x), minus(y, x), z, u), f^#(x, u, z, u)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ perfectp^#(s(x)) -> c_1(f^#(x, s(0()), s(x), s(x)))
, f^#(s(x), 0(), z, u) -> c_2(f^#(x, u, minus(z, s(x)), u))
, f^#(s(x), s(y), z, u) -> c_3(x, y, f^#(x, u, z, u)) }
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: perfectp^#(s(x)) -> c_1(f^#(x, s(0()), s(x), s(x))) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {3}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [0]
[s](x1) = [1] x1 + [0]
[minus](x1, x2) = [0]
[perfectp^#](x1) = [4] x1 + [4]
[f^#](x1, x2, x3, x4) = [0]
[c_1](x1) = [1] x1 + [1]
[c_2](x1) = [4] x1 + [0]
[c_3](x1, x2, x3) = [1] x3 + [0]
The order satisfies the following ordering constraints:
[perfectp^#(s(x))] = [4] x + [4]
> [1]
= [c_1(f^#(x, s(0()), s(x), s(x)))]
[f^#(s(x), 0(), z, u)] = [0]
>= [0]
= [c_2(f^#(x, u, minus(z, s(x)), u))]
[f^#(s(x), s(y), z, u)] = [0]
>= [0]
= [c_3(x, y, f^#(x, u, z, u))]
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:
{ f^#(s(x), 0(), z, u) -> c_2(f^#(x, u, minus(z, s(x)), u))
, f^#(s(x), s(y), z, u) -> c_3(x, y, f^#(x, u, z, u)) }
Weak DPs: { perfectp^#(s(x)) -> c_1(f^#(x, s(0()), s(x), s(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:
{ 2: f^#(s(x), s(y), z, u) -> c_3(x, y, f^#(x, u, z, u))
, 3: perfectp^#(s(x)) -> c_1(f^#(x, s(0()), s(x), s(x))) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {3}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [0]
[s](x1) = [1] x1 + [4]
[minus](x1, x2) = [0]
[perfectp^#](x1) = [2] x1 + [6]
[f^#](x1, x2, x3, x4) = [1] x1 + [5]
[c_1](x1) = [1] x1 + [5]
[c_2](x1) = [1] x1 + [4]
[c_3](x1, x2, x3) = [1] x3 + [0]
The order satisfies the following ordering constraints:
[perfectp^#(s(x))] = [2] x + [14]
> [1] x + [10]
= [c_1(f^#(x, s(0()), s(x), s(x)))]
[f^#(s(x), 0(), z, u)] = [1] x + [9]
>= [1] x + [9]
= [c_2(f^#(x, u, minus(z, s(x)), u))]
[f^#(s(x), s(y), z, u)] = [1] x + [9]
> [1] x + [5]
= [c_3(x, y, f^#(x, u, z, u))]
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:
{ f^#(s(x), 0(), z, u) -> c_2(f^#(x, u, minus(z, s(x)), u)) }
Weak DPs:
{ perfectp^#(s(x)) -> c_1(f^#(x, s(0()), s(x), s(x)))
, f^#(s(x), s(y), z, u) -> c_3(x, y, f^#(x, u, z, u)) }
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: f^#(s(x), 0(), z, u) -> c_2(f^#(x, u, minus(z, s(x)), u))
, 2: perfectp^#(s(x)) -> c_1(f^#(x, s(0()), s(x), s(x)))
, 3: f^#(s(x), s(y), z, u) -> c_3(x, y, f^#(x, u, z, u)) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {3}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [0]
[s](x1) = [1] x1 + [4]
[minus](x1, x2) = [0]
[perfectp^#](x1) = [1] x1 + [1]
[f^#](x1, x2, x3, x4) = [1] x1 + [0]
[c_1](x1) = [1] x1 + [0]
[c_2](x1) = [1] x1 + [1]
[c_3](x1, x2, x3) = [1] x3 + [0]
The order satisfies the following ordering constraints:
[perfectp^#(s(x))] = [1] x + [5]
> [1] x + [0]
= [c_1(f^#(x, s(0()), s(x), s(x)))]
[f^#(s(x), 0(), z, u)] = [1] x + [4]
> [1] x + [1]
= [c_2(f^#(x, u, minus(z, s(x)), u))]
[f^#(s(x), s(y), z, u)] = [1] x + [4]
> [1] x + [0]
= [c_3(x, y, f^#(x, u, z, u))]
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:
{ perfectp^#(s(x)) -> c_1(f^#(x, s(0()), s(x), s(x)))
, f^#(s(x), 0(), z, u) -> c_2(f^#(x, u, minus(z, s(x)), u))
, f^#(s(x), s(y), z, u) -> c_3(x, y, f^#(x, u, z, u)) }
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.
{ perfectp^#(s(x)) -> c_1(f^#(x, s(0()), s(x), s(x)))
, f^#(s(x), 0(), z, u) -> c_2(f^#(x, u, minus(z, s(x)), u))
, f^#(s(x), s(y), z, u) -> c_3(x, y, f^#(x, u, z, u)) }
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))