We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ f(g(x), s(0()), y) -> f(g(s(0())), y, g(x))
, g(s(x)) -> s(g(x))
, g(0()) -> 0() }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ f^#(g(x), s(0()), y) -> c_1(f^#(g(s(0())), y, g(x)))
, g^#(s(x)) -> c_2(g^#(x))
, g^#(0()) -> c_3() }
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:
{ f^#(g(x), s(0()), y) -> c_1(f^#(g(s(0())), y, g(x)))
, g^#(s(x)) -> c_2(g^#(x))
, g^#(0()) -> c_3() }
Strict Trs:
{ f(g(x), s(0()), y) -> f(g(s(0())), y, g(x))
, g(s(x)) -> s(g(x))
, g(0()) -> 0() }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We replace rewrite rules by usable rules:
Strict Usable Rules:
{ g(s(x)) -> s(g(x))
, g(0()) -> 0() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ f^#(g(x), s(0()), y) -> c_1(f^#(g(s(0())), y, g(x)))
, g^#(s(x)) -> c_2(g^#(x))
, g^#(0()) -> c_3() }
Strict Trs:
{ g(s(x)) -> s(g(x))
, g(0()) -> 0() }
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(s) = {1}, Uargs(f^#) = {1, 2, 3}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[g](x1) = [0 1] x1 + [0]
[0 1] [0]
[s](x1) = [1 0] x1 + [0]
[0 1] [1]
[0] = [0]
[1]
[f^#](x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[g^#](x1) = [0]
[0]
[c_2](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_3] = [0]
[0]
The order satisfies the following ordering constraints:
[g(s(x))] = [0 1] x + [1]
[0 1] [1]
> [0 1] x + [0]
[0 1] [1]
= [s(g(x))]
[g(0())] = [1]
[1]
> [0]
[1]
= [0()]
[f^#(g(x), s(0()), y)] = [0 1] x + [1 0] y + [0]
[0 0] [0 0] [0]
? [0 1] x + [1 0] y + [2]
[0 0] [0 0] [0]
= [c_1(f^#(g(s(0())), y, g(x)))]
[g^#(s(x))] = [0]
[0]
>= [0]
[0]
= [c_2(g^#(x))]
[g^#(0())] = [0]
[0]
>= [0]
[0]
= [c_3()]
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:
{ f^#(g(x), s(0()), y) -> c_1(f^#(g(s(0())), y, g(x)))
, g^#(s(x)) -> c_2(g^#(x))
, g^#(0()) -> c_3() }
Weak Trs:
{ g(s(x)) -> s(g(x))
, g(0()) -> 0() }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
Consider the dependency graph:
1: f^#(g(x), s(0()), y) -> c_1(f^#(g(s(0())), y, g(x)))
-->_1 f^#(g(x), s(0()), y) -> c_1(f^#(g(s(0())), y, g(x))) :1
2: g^#(s(x)) -> c_2(g^#(x))
-->_1 g^#(0()) -> c_3() :3
-->_1 g^#(s(x)) -> c_2(g^#(x)) :2
3: g^#(0()) -> c_3()
Only the nodes {2,3} are reachable from nodes {2,3} that start
derivation from marked basic terms. The nodes not reachable are
removed from the problem.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ g^#(s(x)) -> c_2(g^#(x))
, g^#(0()) -> c_3() }
Weak Trs:
{ g(s(x)) -> s(g(x))
, g(0()) -> 0() }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We estimate the number of application of {2} by applications of
Pre({2}) = {1}. Here rules are labeled as follows:
DPs:
{ 1: g^#(s(x)) -> c_2(g^#(x))
, 2: g^#(0()) -> c_3() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { g^#(s(x)) -> c_2(g^#(x)) }
Weak DPs: { g^#(0()) -> c_3() }
Weak Trs:
{ g(s(x)) -> s(g(x))
, g(0()) -> 0() }
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.
{ g^#(0()) -> c_3() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { g^#(s(x)) -> c_2(g^#(x)) }
Weak Trs:
{ g(s(x)) -> s(g(x))
, g(0()) -> 0() }
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: { g^#(s(x)) -> c_2(g^#(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: g^#(s(x)) -> c_2(g^#(x)) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[s](x1) = [1] x1 + [2]
[g^#](x1) = [4] x1 + [0]
[c_2](x1) = [1] x1 + [1]
The order satisfies the following ordering constraints:
[g^#(s(x))] = [4] x + [8]
> [4] x + [1]
= [c_2(g^#(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: { g^#(s(x)) -> c_2(g^#(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.
{ g^#(s(x)) -> c_2(g^#(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))