YES(O(1),O(n^1))
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, -(0(), y) -> 0()
, p(0()) -> 0()
, p(s(x)) -> x }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following innermost weak dependency pairs:
Strict DPs:
{ -^#(x, 0()) -> c_1()
, -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
, -^#(0(), y) -> c_3()
, p^#(0()) -> c_4()
, p^#(s(x)) -> c_5() }
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:
{ -^#(x, 0()) -> c_1()
, -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
, -^#(0(), y) -> c_3()
, p^#(0()) -> c_4()
, p^#(s(x)) -> c_5() }
Strict Trs:
{ -(x, 0()) -> x
, -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
, -(0(), y) -> 0()
, p(0()) -> 0()
, p(s(x)) -> x }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We replace rewrite rules by usable rules:
Strict Usable Rules:
{ p(0()) -> 0()
, p(s(x)) -> x }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ -^#(x, 0()) -> c_1()
, -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
, -^#(0(), y) -> c_3()
, p^#(0()) -> c_4()
, p^#(s(x)) -> c_5() }
Strict Trs:
{ p(0()) -> 0()
, p(s(x)) -> x }
Obligation:
innermost 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(-^#) = {2}, Uargs(c_2) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[0] = [0]
[0]
[s](x1) = [1 0] x1 + [0]
[0 1] [0]
[p](x1) = [1 0] x1 + [2]
[0 1] [0]
[-^#](x1, x2) = [2 1] x1 + [2 0] x2 + [0]
[2 1] [0 0] [0]
[c_1] = [1]
[1]
[c_2](x1) = [1 0] x1 + [2]
[0 1] [2]
[c_3] = [1]
[1]
[p^#](x1) = [1 2] x1 + [2]
[1 2] [2]
[c_4] = [1]
[1]
[c_5] = [1]
[1]
The following symbols are considered usable
{p, -^#, p^#}
The order satisfies the following ordering constraints:
[p(0())] = [2]
[0]
> [0]
[0]
= [0()]
[p(s(x))] = [1 0] x + [2]
[0 1] [0]
> [1 0] x + [0]
[0 1] [0]
= [x]
[-^#(x, 0())] = [2 1] x + [0]
[2 1] [0]
? [1]
[1]
= [c_1()]
[-^#(x, s(y))] = [2 0] y + [2 1] x + [0]
[0 0] [2 1] [0]
? [2 0] y + [2 1] x + [6]
[0 0] [2 1] [2]
= [c_2(-^#(x, p(s(y))))]
[-^#(0(), y)] = [2 0] y + [0]
[0 0] [0]
? [1]
[1]
= [c_3()]
[p^#(0())] = [2]
[2]
> [1]
[1]
= [c_4()]
[p^#(s(x))] = [1 2] x + [2]
[1 2] [2]
> [1]
[1]
= [c_5()]
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:
{ -^#(x, 0()) -> c_1()
, -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
, -^#(0(), y) -> c_3() }
Weak DPs:
{ p^#(0()) -> c_4()
, p^#(s(x)) -> c_5() }
Weak Trs:
{ p(0()) -> 0()
, p(s(x)) -> x }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We estimate the number of application of {1,3} by applications of
Pre({1,3}) = {2}. Here rules are labeled as follows:
DPs:
{ 1: -^#(x, 0()) -> c_1()
, 2: -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
, 3: -^#(0(), y) -> c_3()
, 4: p^#(0()) -> c_4()
, 5: p^#(s(x)) -> c_5() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { -^#(x, s(y)) -> c_2(-^#(x, p(s(y)))) }
Weak DPs:
{ -^#(x, 0()) -> c_1()
, -^#(0(), y) -> c_3()
, p^#(0()) -> c_4()
, p^#(s(x)) -> c_5() }
Weak Trs:
{ p(0()) -> 0()
, p(s(x)) -> x }
Obligation:
innermost 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.
{ -^#(x, 0()) -> c_1()
, -^#(0(), y) -> c_3()
, p^#(0()) -> c_4()
, p^#(s(x)) -> c_5() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { -^#(x, s(y)) -> c_2(-^#(x, p(s(y)))) }
Weak Trs:
{ p(0()) -> 0()
, p(s(x)) -> x }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 3' to
orient following rules strictly.
DPs:
{ 1: -^#(x, s(y)) -> c_2(-^#(x, p(s(y)))) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA) and not(IDA(1)).
[7 7 0] [7 7 0] [0]
[-](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0]
[0] = [0]
[0]
[1 0 0] [2]
[s](x1) = [1 0 0] x1 + [0]
[0 1 1] [0]
[1 0 0] [1 0 0] [1 0 0] [0]
[if](x1, x2, x3) = [1 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
[0 1 1] [0 0 1] [0 0 1] [0]
[1 0 0] [1 0 0] [0]
[greater](x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 0] [0 1 0] [0]
[0 1 0] [0]
[p](x1) = [0 0 2] x1 + [0]
[0 0 2] [0]
[4 0 0] [0]
[-^#](x1, x2) = [0 0 0] x2 + [4]
[4 0 0] [0]
[0]
[c_1] = [0]
[0]
[1 0 0] [5]
[c_2](x1) = [0 0 0] x1 + [3]
[0 0 0] [7]
[0]
[c_3] = [0]
[0]
[7 7 0] [0]
[p^#](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0]
[c_4] = [0]
[0]
[0]
[c_5] = [0]
[0]
The following symbols are considered usable
{p, -^#}
The order satisfies the following ordering constraints:
[p(0())] = [0]
[0]
[0]
>= [0]
[0]
[0]
= [0()]
[p(s(x))] = [1 0 0] [0]
[0 2 2] x + [0]
[0 2 2] [0]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= [x]
[-^#(x, s(y))] = [4 0 0] [8]
[0 0 0] y + [4]
[4 0 0] [8]
> [4 0 0] [5]
[0 0 0] y + [3]
[0 0 0] [7]
= [c_2(-^#(x, p(s(y))))]
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, s(y)) -> c_2(-^#(x, p(s(y)))) }
Weak Trs:
{ p(0()) -> 0()
, p(s(x)) -> x }
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, s(y)) -> c_2(-^#(x, p(s(y)))) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs:
{ p(0()) -> 0()
, p(s(x)) -> x }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(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(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^1))