We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ g(X) -> u(h(X), h(X), X)
, u(d(), c(Y), X) -> k(Y)
, h(d()) -> c(a())
, h(d()) -> c(b())
, f(k(a()), k(b()), X) -> f(X, X, X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ g^#(X) -> c_1(u^#(h(X), h(X), X))
, u^#(d(), c(Y), X) -> c_2(Y)
, h^#(d()) -> c_3()
, h^#(d()) -> c_4()
, f^#(k(a()), k(b()), X) -> c_5(f^#(X, 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:
{ g^#(X) -> c_1(u^#(h(X), h(X), X))
, u^#(d(), c(Y), X) -> c_2(Y)
, h^#(d()) -> c_3()
, h^#(d()) -> c_4()
, f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
Strict Trs:
{ g(X) -> u(h(X), h(X), X)
, u(d(), c(Y), X) -> k(Y)
, h(d()) -> c(a())
, h(d()) -> c(b())
, f(k(a()), k(b()), X) -> f(X, X, X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We replace rewrite rules by usable rules:
Strict Usable Rules:
{ h(d()) -> c(a())
, h(d()) -> c(b()) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ g^#(X) -> c_1(u^#(h(X), h(X), X))
, u^#(d(), c(Y), X) -> c_2(Y)
, h^#(d()) -> c_3()
, h^#(d()) -> c_4()
, f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
Strict Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b()) }
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_1) = {1}, Uargs(u^#) = {1, 2}
TcT has computed the following constructor-restricted matrix
interpretation.
[h](x1) = [2]
[0]
[d] = [0]
[0]
[c](x1) = [0]
[0]
[k](x1) = [0]
[0]
[a] = [0]
[0]
[b] = [0]
[0]
[g^#](x1) = [2 2] x1 + [0]
[0 0] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[u^#](x1, x2, x3) = [1 0] x1 + [2 0] x2 + [0]
[0 0] [0 0] [0]
[c_2](x1) = [0]
[0]
[h^#](x1) = [0]
[0]
[c_3] = [0]
[0]
[c_4] = [0]
[0]
[f^#](x1, x2, x3) = [0]
[0]
[c_5](x1) = [0]
[0]
The order satisfies the following ordering constraints:
[h(d())] = [2]
[0]
> [0]
[0]
= [c(a())]
[h(d())] = [2]
[0]
> [0]
[0]
= [c(b())]
[g^#(X)] = [2 2] X + [0]
[0 0] [0]
? [6]
[0]
= [c_1(u^#(h(X), h(X), X))]
[u^#(d(), c(Y), X)] = [0]
[0]
>= [0]
[0]
= [c_2(Y)]
[h^#(d())] = [0]
[0]
>= [0]
[0]
= [c_3()]
[h^#(d())] = [0]
[0]
>= [0]
[0]
= [c_4()]
[f^#(k(a()), k(b()), X)] = [0]
[0]
>= [0]
[0]
= [c_5(f^#(X, 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:
{ g^#(X) -> c_1(u^#(h(X), h(X), X))
, u^#(d(), c(Y), X) -> c_2(Y)
, h^#(d()) -> c_3()
, h^#(d()) -> c_4()
, f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b()) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We estimate the number of application of {1,3,4,5} by applications
of Pre({1,3,4,5}) = {2}. Here rules are labeled as follows:
DPs:
{ 1: g^#(X) -> c_1(u^#(h(X), h(X), X))
, 2: u^#(d(), c(Y), X) -> c_2(Y)
, 3: h^#(d()) -> c_3()
, 4: h^#(d()) -> c_4()
, 5: f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { u^#(d(), c(Y), X) -> c_2(Y) }
Weak DPs:
{ g^#(X) -> c_1(u^#(h(X), h(X), X))
, h^#(d()) -> c_3()
, h^#(d()) -> c_4()
, f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b()) }
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^#(X) -> c_1(u^#(h(X), h(X), X))
, h^#(d()) -> c_3()
, h^#(d()) -> c_4()
, f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { u^#(d(), c(Y), X) -> c_2(Y) }
Weak Trs:
{ h(d()) -> c(a())
, h(d()) -> c(b()) }
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: { u^#(d(), c(Y), X) -> c_2(Y) }
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: u^#(d(), c(Y), X) -> c_2(Y) }
Sub-proof:
----------
The following argument positions are usable:
none
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[d] = [0]
[c](x1) = [0]
[u^#](x1, x2, x3) = [4] x3 + [1]
[c_2](x1) = [0]
The order satisfies the following ordering constraints:
[u^#(d(), c(Y), X)] = [4] X + [1]
> [0]
= [c_2(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: { u^#(d(), c(Y), X) -> c_2(Y) }
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.
{ u^#(d(), c(Y), X) -> c_2(Y) }
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))